IEEE Transactions on Microwave Theory and Techniques – March 2018.

Table of contents :
001 - MTT-066-V003 (2018-03) - COVER_A......Page 1
002 - [email protected] - A Linear Model for Microwave Imaging of Highly Conductive Scatterers......Page 4
003 - [email protected] - Temporal Coupled-Mode Theory of Electromagnetic Components Described by Magnetic Groups of Symmetry......Page 20
004 - [email protected] - A Short–Open Calibration Method for Accurate De-Embedding of 3-D Nonplanar Microstrip Line Structures......Page 27
005 - [email protected] - Modal Analysis and Propagation Characteristics of Leaky Waves on a 2-D Periodic Leaky-Wave Antenna......Page 36
006 - [email protected] - A Scalable Multiharmonic Surface-Potential Model of AlGaN-GaN HEMTs......Page 47
007 - [email protected] - A Passive PEEC-Based Micromodeling Circuit for High-Speed Interconnection Problems......Page 56
008 - [email protected] - Single-Band and Switchable Dual-Single-Band Tunable BPFs With Predefined Tuning Range, Bandwidth, and Selectivity......Page 70
009 - [email protected] - A New Class of K-Band High-Q Frequency-Tunable Circular Cavity Filter......Page 83
010 - [email protected] - Tunable SIW Cavity-Based Dual-Mode Diplexers With Various Single-Ended and Balanced Ports......Page 93
011 - [email protected] - Design Methodologies of Compact Orthomode Transducers Based on Mechanism of Polarization Selectivity......Page 104
012 - [email protected] - Compact Design of Planar Quadrature Coupler With Improved Phase Responses and Wide Tunable Coupling Ratios [Feng Lin]......Page 116
028 - [email protected] - Homodyne Digitally Assisted and Spurious-Free Mixerless Direct Carrier Modulator With High Carrier Leakage Suppression......Page 126
029 - [email protected] - Direct Error-Searching SPSA-Based Model Extraction for Digital Predistortion of RF Power Amplifiers......Page 140
030 - [email protected] - A System-on-Chip Crystal-Less Wireless Sub-GHz Transmitter......Page 152
040 - [email protected] - Digital Predistortion for Multi-Antenna Transmitters Affected by Antenna Crosstalk......Page 161
050 - [email protected] - Exposure Evaluation of an Actual Wireless Power Transfer System for an Electric Vehicle With Near-Field Measurement......Page 173
060 - [email protected] - A Planar Dipole Array Surface for Electromagnetic Energy Harvesting and Wireless Power Transfer......Page 183
070 - [email protected] - A 40-nm CMOS Complex Permittivity Sensing Pixel for Material Characterization at Microwave Frequencies......Page 191
080 - [email protected] - High Power Integrated Photonic W-Band Emitter......Page 207
090 - [email protected] - SensorAntenna Interface IC for Implantable Biomedical Monitoring System......Page 217
100 - [email protected] - Highly Linear and Reconfigurable Three-Way Amplitude Modulation-Based Mixerless Wireless Transmitter......Page 225
110 - [email protected] - A Standing-Wave Architecture for Scalable and Wideband Millimeter-Wave and Terahertz Coherent Radiator Arrays......Page 233
120 - [email protected] - A Fully Parallel Architecture for Designing Frequency-Agile and Real-Time Reconfigurable FPGA-Based RF Digital Transmitters......Page 246
130 - [email protected] - 0.3-THz SiGe-Based High-Efficiency Push–Push VCOs With 1-mW Peak Output Power Employing......Page 257
140 - [email protected] - Inkjet Printing of Epidermal RFID Antennas by Self-Sintering Conductive Ink......Page 272
150 - [email protected] - Polyphase-Basis Discrete Cosine Transform for Real-Time Measurement of Heart Rate With CW Doppler Radar......Page 281
160 - [email protected] - 15 GHz Doherty Power Amplifier With RF Predistortion Linearizer in CMOS SOI......Page 297
170 - [email protected] - Digital Suppression of Transmitter Leakage in FDD RF Transceivers Aliasing Elimination and Model Selection......Page 307
190 - [email protected] - Broadband High-Power W-Band Amplifier MMICs Based on Stacked-HEMT Unit Cells......Page 319
200 - [email protected] - An Extended 4 × 4 Butler Matrix With Enhanced Beam Controllability and Widened Spatial Coverage......Page 326
220 - [email protected] - A 2.33-GHz, −133-dBcHz, and Eight-Phase Oscillator With Dual Tanks and Adaptive Feedback......Page 337
230 - [email protected] - Theory and Experiment of Two-Section Two-Resistor Wilkinson Power Divider With Two Arbitrary Frequency Bands......Page 349
240 - [email protected] - A Novel Method for 3-D Millimeter-Wave Holographic Reconstruction Based on Frequency Interferometry Techniques......Page 359
250 - [email protected] - A CMOS Real-Time Spectrum Sensor Based on Phasers for Cognitive Radios [Paria Sepidband]......Page 377
260 - [email protected] - Quasi-Optical Input Mode Coupler for a Ka-Band Multimegawatt Gyroklystron......Page 389
270 - [email protected] - Simultaneous Imaging, Sensor Tag Localization, and Backscatter Uplink via Synthetic Aperture Radar......Page 395
280 - [email protected] - Integrated Quasi-Circulator With RF Leakage Cancellation for Full-Duplex Wireless Transceivers......Page 404
290 - [email protected] - Analytical Approach for SiGe HBT Static Frequency Divider Design for Millimeter-Wave Frequency Operation......Page 414
350 - [email protected] - An Injection- and Frequency-Locked Loop for Reducing Phase Noise of Wideband Oscillators......Page 421
400 - [email protected] - Design of an 87% Fractional Bandwidth Doherty Power Amplifier Supported by a Simplified Bandwidth Estimation Method......Page 431
410 - [email protected] - A Load Modulated Balanced Amplifier for Telecom Applications [Roberto Quaglia]......Page 440
420 - [email protected] - Compression Point Enhancement by Controlling the Expansion Inherently......Page 451
430 - [email protected] - A Fully Parallel Architecture for Designing Frequency-Agile and Real-Time Reconfigurable FPGA-Based RF Digital Transmitters......Page 461
440 - [email protected] - Stepped-Carrier OFDM-Radar Processing Scheme to Retrieve High-Resolution Range-Velocity Profile at Low Sampling Rate......Page 472
450 - MTT-066-V003 (2018-03) - INSIDE......Page 481
460 - MTT-066-V003 (2018-03) - COVER_B......Page 482
810 - [email protected] - 60-GHz CMOS Doppler Radar Sensor With Integrated V-Band Power Detector for Clutter Monitoring......Page 484
810 - [email protected] - Design Methodology for Six-Port EqualUnequal Quadrature and Rat-Race Couplers......Page 493

Citation preview

IEEE TRANSACTIONS ON

MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY

JR MTT-5

MARCH 2018

VOLUME 66

NUMBER 3

IETMAB

(ISSN 0018-9480)

THIS ISSUE INCLUDES THE JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS REGULAR PAPERS OF THE TRANSACTIONS ON MICRO\\'AVE THEORY AND TECHNIQUES

EM Theory and Analysis Techniques A Linear Model for Microwave Imag ing or Highly Conductive Scatterers . .......................... . ...... ... ............ . . .... . .. . . .. . . .. .. . ........ . ... .. .. ... . . .. .............. ..... ... . .... . ............. .. . ... S. Sun, B . .!. Kooij. and J\. G. Ycum·o,· Temporal Coupled-Mode Theory or E lectromagnetic Components Described by Mag netic Groups o r Symmetry . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V D111itrie1', G. Portela, and L. Ma rtins A Short- Open Calibration Method for Accurate De-Embedding of 3-D Nonplanar Microstrip Line Structures in Finite-Element Method .. ... ... . . ....... . .. . . . ....... . . .. . . ....... . .... ... .. ... ... .... . . ..... . .. ..... ... . . . . . . Y Li and L. Zhu Modal Analysis and Propagation Characteristics of Leaky Waves on a 2-D Periodic Leaky-Wave Antenna .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Srng upta, n. R. Jackson, and S. J\. Long

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Devices and Modeling A Scalable Multiharmonic Surface-Potential Model of A IGaN/GaN HEMTs .... . ..... ... . . .......................... . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Wu, Y Xu, Y Chen, Y Wan g, W Fu. B. Yan. and R. Xu A Passive PEEC-Bascd Micromodcling Circuit fo r High-Speed fnterconnccti on Problems .. ... Y. Dou and K. -L. Wu

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Passive Circuits Single-Band and Switchable Dual-/Single-Band Tunable BPFs With Predefined Tuning Range, Bandwidth. and Selectivity . .... . .. . .... . ... . ..... . ...... . . . ... . ... . ....... .. ... .. ... . ........... .. D. Lu. X. Tang, N. S. Barka and Y Feng A New Class of K-Band High-Q Frequency-Tunab le Circular Cavity Filler .... . . .... .... . ......... . . ................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Nam, B. Lee, C. K1l'ak, and .!. Lee Tunable SIW Cavity-Based Dual -Mode Diplexers With Various Single-Ended and Balanced Ports ....... . ............ . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . M. F Ha?,ag, M. !\Im Khate 1; M. D. Hickle, and D. Peroufis Design Methodology !'or Six-Port Eq ual/Unequal Quadrature and Rat-Race Couplers With Balanced and Unbalanced Ports Terminated by Arbitrary Resistances .. .. ...... .. . .. ...... .. .. ...... ...... . ....... . . ... ... . . . .... . . .. .. .. .. .. . . ... ... .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. .liao, Y. Wu, W. Zhan g. M. Li, Y. Liu, Q. Xue, and Z. Ghassemlooy

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+IEEE

(Contents Continued from Front Cover) Compact Design of Planar Quadrature Coupler With Improved Phase Responses and Wide Tunable Coupling Ratios ........................................................................................................................ F. Lin Quasi-Optical Input Mode Coupler for a Ka-Band Multimegawatt Gyroklystron ........................................... ............. E. B. Abubakirov, Y. M. Guznov, S. V. Kuzikov, A. S. Shevchenko, A. A. Vikharev, and S. A. Zapevalov Design Methodologies of Compact Orthomode Transducers Based on Mechanism of Polarization Selectivity ......... ................................................................................................. A. A. Sakr, W. Dyab, and K. Wu Theory and Experiment of Two-Section Two-Resistor Wilkinson Power Divider With Two Arbitrary Frequency Bands .................................................................................................. X. Wang, Z. Ma, and M. Ohira An Extended 4 × 4 Butler Matrix With Enhanced Beam Controllability and Widened Spatial Coverage ............... .......................................................................................................... H. N. Chu and T.-G. Ma Hybrid and Monolithic RF Integrated Circuits Broadband High-Power W-Band Amplifier MMICs Based on Stacked-HEMT Unit Cells ................................ ................................................................... F. Thome, A. Leuther, M. Schlechtweg, and O. Ambacher Design of an 87% Fractional Bandwidth Doherty Power Amplifier Supported by a Simplified Bandwidth Estimation Method ................................................... J. J. Moreno Rubio, V. Camarchia, M. Pirola, and R. Quaglia A Load Modulated Balanced Amplifier for Telecom Applications ............................ R. Quaglia and S. Cripps 15 GHz Doherty Power Amplifier With RF Predistortion Linearizer in CMOS SOI ....................................... ............................................................................. N. Rostomyan, J. A. Jayamon, and P. M. Asbeck Compression Point Enhancement by Controlling the Expansion Inherently .................................................. ................................................................ E. Sobotta, U. Jörges, R. Wolf, D. Fritsche, and F. Ellinger Design of a D-Band CMOS Amplifier Utilizing Coupled Slow-Wave Coplanar Waveguides ............................. ................................... D. Parveg, M. Varonen, D. Karaca, A. Vahdati, M. Kantanen, and K. A. I. Halonen An Injection- and Frequency-Locked Loop for Reducing Phase Noise of Wideband Oscillators ......................... ......................................................... K.-C. Peng, C.-H. Lee, D.-G. Wong, F.-K. Wang, and T.-S. Horng 0.3-THz SiGe-Based High-Efficiency Push–Push VCOs With >1-mW Peak Output Power Employing Common-Mode Impedance Enhancement ......................................... F. Ahmed, M. Furqan, B. Heinemann, and A. Stelzer A 2.33-GHz, −133-dBc/Hz, and Eight-Phase Oscillator With Dual Tanks and Adaptive Feedback ...................... ............................................................................................... R. Jiang, H. Noori, and F. F. Dai Analytical Approach for SiGe HBT Static Frequency Divider Design for Millimeter-Wave Frequency Operation .... ......................................................................................................... A. Dyskin and I. Kallfass

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JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS JOURNAL WITHIN A JOURNAL PAPERS

Wireless Communication Systems Integrated Quasi-Circulator With RF Leakage Cancellation for Full-Duplex Wireless Transceivers ...................... ......................................................................... S. A. Ayati, D. Mandal, B. Bakkaloglu, and S. Kiaei A System-on-Chip Crystal-Less Wireless Sub-GHz Transmitter ............................................................... ........................... P. Greiner, J. Grosinger, J. Schweighofer, C. Steffan, S. Wilfling, G. Holweg, and W. Bösch A CMOS Real-Time Spectrum Sensor Based on Phasers for Cognitive Radios ........ P. Sepidband and K. Entesari Millimeter-Wave Multifunction Multiport Interferometric Receiver for Future Wireless Systems ........................ ........................................................................................................ J. Moghaddasi and K. Wu Highly Linear and Reconfigurable Three-Way Amplitude Modulation-Based Mixerless Wireless Transmitter ......... .................................................................................................. S. Illath Veetil and M. Helaoui Homodyne Digitally Assisted and Spurious-Free Mixerless Direct Carrier Modulator With High Carrier Leakage Suppression ........................ W. Zhang, A. Hasan, F. M. Ghannouchi, M. Helaoui, Y. Wu, L. Jiao, and Y. Liu A Fully Parallel Architecture for Designing Frequency-Agile and Real-Time Reconfigurable FPGA-Based RF Digital Transmitters ................................. D. C. Dinis, R. F. Cordeiro, A. S. R. Oliveira, J. Vieira, and T. O. Silva Digital Suppression of Transmitter Leakage in FDD RF Transceivers: Aliasing Elimination and Model Selection .... ........................................................................................................ W. Cao, Y. Li, and A. Zhu Direct Error-Searching SPSA-Based Model Extraction for Digital Predistortion of RF Power Amplifiers .............. ............................................................................................................... N. Kelly and A. Zhu Digital Predistortion for Multi-Antenna Transmitters Affected by Antenna Crosstalk ...................................... .................................................... K. Hausmair, P. N. Landin, U. Gustavsson, C. Fager, and T. Eriksson

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(Contents Continued from Page 1147) Wireless Power Transfer and RFID Systems Coupling Matrix Synthesis and Impedance-Matching Optimization Method for Magnetic Resonance Coupling Systems ................................................................................... K. A. Thackston, H. Mei, and P. P. Irazoqui Exposure Evaluation of an Actual Wireless Power Transfer System for an Electric Vehicle With Near-Field Measurement ............................................. J. Chakarothai, K. Wake, T. Arima, S. Watanabe, and T. Uno A Planar Dipole Array Surface for Electromagnetic Energy Harvesting and Wireless Power Transfer .................. ............................................................................. A. Z. Ashoor, T. S. Almoneef, and O. M. Ramahi Inkjet Printing of Epidermal RFID Antennas by Self-Sintering Conductive Ink ............................................ ................................................................................... S. Amendola, A. Palombi, and G. Marrocco Simultaneous Imaging, Sensor Tag Localization, and Backscatter Uplink via Synthetic Aperture Radar ................ ...................................... X. Fu, A. Pedross-Engel, D. Arnitz, C. M. Watts, A. Sharma, and M. S. Reynolds Microwave Imaging and Radar Applications A Novel Method for 3-D Millimeter-Wave Holographic Reconstruction Based on Frequency Interferometry Techniques ................................................................................. J. Gao, Y. Qin, B. Deng, H. Wang, and X. Li A Standing-Wave Architecture for Scalable and Wideband Millimeter-Wave and Terahertz Coherent Radiator Arrays .......................................................................................................... H. Jalili and O. Momeni Stepped-Carrier OFDM-Radar Processing Scheme to Retrieve High-Resolution Range-Velocity Profile at Low Sampling Rate ................................................ B. Schweizer, C. Knill, D. Schindler, and C. Waldschmidt Microwave Sensors and Biomedical Applications A 40-nm CMOS Complex Permittivity Sensing Pixel for Material Characterization at Microwave Frequencies ....... ................................................ G. Vlachogiannakis, M. A. P. Pertijs, M. Spirito, and L. C. N. de Vreede 60-GHz CMOS Doppler Radar Sensor With Integrated V-Band Power Detector for Clutter Monitoring and Automatic Clutter-Cancellation in Noncontact Vital-Signs Sensing ..................................................................... ..................................................................... C.-C. Chou, W.-C. Lai, Y.-K. Hsiao, and H.-R. Chuang Polyphase-Basis Discrete Cosine Transform for Real-Time Measurement of Heart Rate With CW Doppler Radar ... ........................................... J. Park, J.-W. Ham, S. Park, D.-H. Kim, S.-J. Park, H. Kang, and S.-O. Park Sensor/Antenna Interface IC for Implantable Biomedical Monitoring System ............................................... ............................................................................................ J.-Y. Lin, H.-C. Chen, and M.-Y. Yen Microwave Photonics High Power Integrated Photonic W-Band Emitter .............. K. Sun, J. Moody, Q. Li, S. M. Bowers, and A. Beling

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A Linear Model for Microwave Imaging of Highly Conductive Scatterers Shilong Sun , Bert Jan Kooij, and Alexander G. Yarovoy, Fellow, IEEE Abstract— In this paper, a linear model based on multiple measurement vectors’ model is proposed to formulate the inverse scattering problem of highly conductive objects at one single frequency. Considering the induced currents that are mostly distributed on the boundaries of the scatterers, joint sparse structure is enforced by a sum-of-norm regularization. Since no a priori information is required and no approximation of the scattering model has been made, the proposed method is versatile. Imaging results with transverse magnetic and transverse electric polarized synthetic data and Fresnel data demonstrate its higher resolving ability than both linear sampling method and its improved version with higher, but acceptable, computational complexity. Index Terms— Inverse scattering problem, joint sparse structure, multiple measurement vectors (MMVs), sum-of-norm regularization constraint, transverse electric (TE), transverse magnetic (TM).

I. I NTRODUCTION

I

NVERSE scattering is a procedure of recovering the characteristics of the objects from the scattered fields. It is of great importance because of its wide applications in different areas, such as seismic detection, medical imaging, sonar, remote sensing, and so forth. Most of the studies on the inverse scattering problems are focused on the frequencies of the resonant region, i.e., the wavelength is comparable with the dimension of the object. Challenges mainly lie in the nonlinearity and ill-posedness in the Hadamard sense [1]. There is a large variety of possible inverse scattering problems, for example, find the shape of the scatterer with the boundary condition already known, or find the space dependent coefficients of the object without any a priori information at all. The inverse scattering problem discussed in this paper is to determine the shape of the highly conductive scatterers with the scattered electromagnetic (EM) field for one or several incident fields at one single frequency of the resonant region. Basically, there are two families of methods for solving this problem: the volume-based methods and the surfacebased methods. The volume-based methods indicate the shape with space-dependent coefficients. Kleinman and den Berg [2] proposed to retrieve the boundary of the highly conductive scatterer by doing the same with the iterative method of reconstructing the conductivity of an EM penetrable object. Manuscript received July 17, 2017; revised September 25, 2017; accepted November 1, 2017. Date of publication November 28, 2017; date of current version March 5, 2018. (Corresponding author: Shilong Sun.) The authors are with the Group of Microwave Sensing, Signals and Systems, Delft University of Technology, 2628 Delft, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2772795

This idea was further extended to the mixed dielectric and highly conductive objects combined with contrast source inversion (CSI) method [3] (see [4], [5]). Classical design of the cost functional consists of the data error and the state error without considering the cross-correlated mismatch of both errors. Recently, a so-called cross-correlated error function and a novel inversion method, referred to as cross-correlated CSI, have been proposed to enhance the inversion ability [6]. The idea of solving the nonlinear inverse scattering problem with linear algebra can also be found in [7]. An algorithmically efficient algorithm, the time-reversal multiple signal classification (TR-MUSIC) method [8], [9], is also of interest since it is capable to break through the diffraction limit. Linear sampling method (LSM) [10]–[12] is another typical volume-based method, which finds an indicator function for each voxel in the region of interest by first defining a far-field mapping operator (or a near-field mapping operator [13]) and then solving a linear system of equations. The norm of the indicator function approaches to zero when the position of the corresponding voxel approaches the highly conductive scatterer. Although LSM has been proved to be effective for highly conductive scatterer, and also applicable for dielectric scatterer in some cases [14], this method needs sufficient amount of measurements to guarantee the inversion performance [15]. Besides, it is very time consuming to compute the dyadic Green function for an irregular inhomogeneous background [16], for instance, in the case of ground penetrating radar [17]. The surfacebased methods first parameterize the shape of the scatterer mathematically with several parameters, and then optimize the parameters by minimizing a cost function iteratively [18]. The drawback of this method is obvious. First, this type of method requires a priori information on the position and quantity of the scatterers; more research on this point can be found in [19] and [20]. Second, it is intractable to deal with the complicated nonconvex objects. Apart from that, each iteration involves finding a solution to a forward scattering problem, which is extremely time consuming for the large-scale inverse problems with an irregular background. As a matter of fact, this is a general drawback of the iterative inversion methods. In cases where the dimension of the solution space is not so huge, global optimization techniques [21]–[23] are good candidates to search for the global optimal solution. We also refer to [24] for a compressive sensing CSI method that solves the contrast source two-step formulation for detecting the nonradiating part of the equivalent currents. Recently, we have proposed a linear model to address the nonlinear highly conductive inverse scattering problem with transverse magnetic (TM) polarized incident fields [25].

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 3, MARCH 2018

The basic idea is to transfer the problem to a set of linear inverse source problems, formulate the set of problems with the multiple measurement vector (MMV) model [26], and finally solve the problem with the sum-of-norm regularization constraint. We have also considered a cascade of the linear inverse source model and a linear optimization model for solving the 3-D inverse scattering problem with half-space configurations [27]. Although the feasibility of this idea has been demonstrated numerically, a theoretical framework has not been established yet. The extension in solving the inverse scattering problem of vectorial fields is not straightforward and therefore has to be proved theoretically. Moreover, the feasibility needs to be further demonstrated with experimental measurement data. In this paper, we first presented the theoretical framework in solving the highly conductive inverse scattering problem with TM-polarized incident fields. Based on the convex optimization theory and the spectral projected gradient (SPG) method (SPGL1 ) [28], [29], we extended the theoretical framework to solve the transverse electric (TE) polarized (vectorial field) inverse scattering problems. Consequently, the extension in 3-D problems can be derived straightforwardly. Cross-validation (CV) technique [30], [31] is used to terminate the iteration such that the estimation of noise level is well circumvented. Both 2-D synthetic data generated by a MATLAB-based “MaxwellFDFD” package [32] and the experimental TM- and TE-polarized data sets of the Institut Fresnel, Marseille, France [33], are processed for demonstrating the validity of the proposed model. What is more, we have also presented an analysis on the computational complexity and the effect of transmitter/receiver number on the imaging quality of the proposed approach, which sheds a light on the design of the imaging system. For the case of penetrable objects, the contrast sources are distributed everywhere in the interior of the object. Since the linear model is regularized with sum-of-norm constraint, the reconstruction algorithm tends to seek for a solution of the minimum sum of norm. According to the field equivalence principle [34], for penetrable object, there would more likely be several sparse solutions that not only generate the same scattered field pattern (satisfying the data equation) but also have the same nonzero structure (possessing smaller sum of norm than the real solutions). Since correct recovery cannot be guaranteed for penetrable objects, we restricted the discussion in this paper to highly conductive objects. In a nutshell, major differences of the proposed approach in comparison with other existing methods are as follows. 1) In comparison with linear methods with linear model of weak scattering assumptions (such as Born/Rytov approximations), the proposed model is more applicable since no weak scattering approximation has been made. 2) In comparison with linear methods with linear model of no weak scattering approximations (such as LSM), the joint sparse structure of the contrast sources is enforced in the proposed approach by the use of sum-of-norm regularization constraint, resulting in higher resolving ability. 3) In comparison with linear iterative algebra of multiple levels with nonlinear model (such as CSI), calculation of

Fig. 1. Configuration of the inverse scattering problem with respect to highly conductive scatterers.

the total fields for the proposed approach is not needed, resulting in higher imaging efficiency. 4) In comparison with super resolution methods based on pseudospectrum analysis (such as TR-MUSIC), the proposed approach does not need to estimate the scatterer number nor does it need to care about how the imaging domain is discretized. Since the proposed approach is also based on a linear model of no weak scattering approximation, LSM and an improved version are selected in this paper for comparison. The remainder of this paper is organized as follows. In Section II-A, the problem statement is given. In Section II-B, the formulation of the linear model is presented. In Section II-C, the SPGL1 method for solving the single measurement vector (SMV) model in TM case is introduced. In Sections II-D and II-E, we derived the sumof-norm optimization method for solving the MMV model of TM case and TE case, respectively. In Section II-F, a CV-based modified SPGL1 method is introduced. In Section III, LSM and its improved version are introduced. The inverted results with synthetic data and experimental data are given in Sections IV and V, respectively. Finally, Section VI ends this paper with our conclusions. II. MMV L INEAR I NVERSION M ODEL A. Problem Statement We consider a scattering configuration as depicted in Fig. 1, which consists of a bounded simply connected inhomogeneous background domain D. The domain D contains a highly conductive cylinder , whose surface is represented by ∂. The dielectric properties of the background are known beforehand. The domain S contains the sources and receivers. The sources are denoted by the subscript p (where p ∈ {1, 2, 3 . . . , P}), and the receivers are denoted by the subscript q (where q ∈ {1, 2, 3, . . . , Q}). Sources and receivers that have equal subscripts are located at the same position. We use a righthanded coordinate system in which the unit vector in the invariant direction points out of this paper. In our notation for the vectorial quantities, we use a bold notation that represents a vector with three components.

SUN et al.: LINEAR MODEL FOR MICROWAVE IMAGING OF HIGHLY CONDUCTIVE SCATTERERS

The general mathematical representations presented are consistent with any 3-D configuration, in which the 2-D TE and TM excitations are a special case, resulting in vectors containing zero elements. In this paper, we consider the time factor exp(i ωt). Here, i represents the imaginary unit, and then the vectorial Maxwell’s equations in the frequency domain can be written as      −i ωε − σ ∇× E J = (1) ∇× i ωμ H −M where E and H are the electric and magnetic fields, respectively; J and M are the electric and magnetic current source densities, respectively; and σ , ε, and μ are the electric conductivity, electric permittivity, and magnetic permeability, respectively. For most of the real problems, μ can be reasonably assumed to be the permeability of free space, μ0 , while σ and ε are functions of both the position vector x = [x 1 , x 2 , x 3 ]T and the angular frequency ω. Since the solution of the E-field is two orders of magnitude larger than the H -field, it is for better numerical accuracy (see [32]) to eliminate either the E-field or the H -field from (1). In this paper, we assume the electric field is measured, so we rewrite our equations in terms of the electric field E according to ∇ × μ−1 ∇ × E − ω2  E = −i ω J − ∇ × μ−1 M

(2)

where  is the complex permittivity given by  = ε − i σ /ω.

(3)

The total electric field, E p , and the incident electric field, E inc p , are excited by the pth external source, and the scattered electric field E sct p is then found by inc E sct p = Ep − Ep ,

p = 1, 2, . . . , P.

(4)

Here, the subscript p corresponds to the pth source. According to the above relation and the electric field equation (2), it is easy to obtain the basic equation of the inverse scattering problem, which is sct 2 sct 2 ∇ × μ−1 0 ∇ × E p − ω b E p = ω χ E p ,

p = 1, 2, . . . , P (5)

where the contrast χ is the difference of the complex permittivity of the test domain, , and that of the background, b , i.e., χ =  − b . The problem we are going to resolve is to find the shape of the highly conductive scatterer ∂ from the measurement of the scattered electric fields E sct p . Since the total electric field E p is a function of the contrast χ , this is obviously a nonlinear problem.

1151

where A is the stiffness matrix, and ⎡ sct ⎤ ⎡ ⎤ e p,x1 e p,x1 ⎥ ⎢ sct e p = ⎣e p,x2 ⎦ esct p = ⎣ e p,x 2 ⎦ e p,x3 esct p,x 3

(7)

and χ e p is the component-wise multiplication of the two −1 2 vectors, χ and e p . The scattered fields esct p = A ω χ ep are probed and the measurements are used to estimate the unknown χ . Now let us use a measurement operator, MS , to select the field values at the receiver positions, and then we can formulate the data equations as y p = MS A−1 ω2 χ e p , where y p is the measurement vector of the pth scattered fields. Let us further define the contrast source as j p := χ e p , and then we have yp =  j p ,

p = 1, 2, 3, . . . , P

(8)

where  is the sensing matrix defined by  = MS A−1 ω2 .

(9)

In TM case,  ∈ C Q×N , while in TE case,  ∈ C2Q×2N . Here, N is the grid number of the discretized inversion domain. Since the contrast source j p shows sparsity due to the fact that the induced current exists only on the surface of the highly conductive objects, the ill-posedness of the inverse scattering problem can be overcome by exploiting the sparsity of the contrast sources. Further, although the contrast sources j p excited by the illumination of the incident fields einc p are of different values, the nonzero elements are located at the same positions—the boundary of the highly conductive scatterers. This inspired us to improve the inversion performance by enhancing the joint sparse structure, so the linear data model (8) is further formulated as an MMV model. In the following section, a linear model is constructed and a sum-of-norm optimization problem is derived for TM and TE cases, respectively. In doing so, the nonlinear inverse scattering problem can be simplified and addressed by a linear optimization scheme without considering the state equations (i.e., the calculation of the total fields is circumvented) −1 2 e p = esct p + A ω jp,

p = 1, 2, . . . , P.

(10)

Specifically, (8) is rewritten as follows: Y = J + U

(11)

where  is the joint sensing matrix J = [ j1 , j2 , . . . , j P ]

B. Formulation

is the contrast source matrix, and

First, let us formulate the inverse problem following the vector form of the FDFD scheme in [32], and rewrite (5) as follows: 2 Aesct p = ω χ e p,

p = 1, 2, 3, . . . , P

(6)

Y = [ y1 , y2 , . . . , yP ] is the measurement data matrix, and U represents the additive complex measurement noise matrix.

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C. Solving the SMV Model: TM Case First, consider the single source configuration illuminated by TM-polarized wave. The inverse scattering problem is formulated as a basis pursuit denoise (BP

σ ) problem [35] σ (SMVTM BP

σ ) : min  j p 1 s.t.  j p − y p 2 ≤

(12)

where

σ represents the noise level, and the contrast source is regularized with the 1 -norm constraint. Solving this problem means searching for a solution j p that is of the smallest 1 -norm and meanwhile satisfies the inequality condition. Although the BP

σ problem is straightforward for understanding the inverse problem, it is not easy to solve directly even if we exactly know the value of

σ . An equivalent problem that is much simpler to solve is the Lasso (LSτ ) problem [36], which is formulated as

Fig. 2.

Probing the Pareto curve: the update of parameter τ .

(SMVTM LSτ ) : min  j p − y p 2 s.t.  j p 1 ≤ τ. (13) The LSτ problem can be solved using an SPG method that is proposed based on convex optimization theory [37]–[39]. Details of the SPG method for solving the (LSτ ) problem (13) are given in [28, Algorithm 1], in which Pτ [·] is a projection operator defined as  (14) Pτ [ j p ] := arg min  j p − s2 s.t. s1 ≤ τ . s

Pτ [·] gives the projection of a vector j p onto the one-norm ball with radius τ . In practice, τ is usually not available. For solving this problem, a Pareto curve is defined in SPGL1 algorithm [28] by φSMVTM (τ ) =  j p,τ − y p 2

(15)

where j p,τ is the optimal solution to the LSτ problem. It is easy to find that the (BP

σ ) problem is equivalent to the (LSτ ) problem when φSMVTM (τ ) =

σ is satisfied. The Pareto curve is proved to be differentiable under some conditions, and the σ can be reached root of the nonlinear equation φSMVTM (τ ) =

by Newton iterations [28] τh+1 = τh +

σ − φSMVTM (τh )  φSMV (τh ) TM

(16)

where  φSMV (τh+1 ) = − TM

 H r

p,τh ∞

r p,τ 2

(17)

H

where is the conjugate transpose of matrix  and r p,τh =  j p,τh − y p is the residual vector. The update of τ by probing the Pareto curve is illustrated in Fig. 2. This procedure requires computing successively more accurate solutions of LSτ . The Newton root-finding framework for solving the (SMVTM BP

σ) problem is given in [29, Algorithm 1]. D. Solving the MMV Model: TM Case Let us now first consider the 2-D multisource configuration with TM-polarized illumination. As all the contrast sources are focused on the boundary of the scatterers, the contrast source matrix J shows row sparsity. Therefore, the inverse scattering

problem with multisource configurations can be formulated as an (MMVTM BP

σ ) problem regularized by a sum-of-norm constraint (MMVTM BP

σ σ ) min κ( J ) s.t.  J − Y  F ≤

where κ( J ) is the mixed (α, β)-norm defined as 1/α

N   α T J   J α,β :=

(18)

(19)

i,: β

i=1

with Ji,: denoting the i th row of J , and ·β the conventional β-norm.  ·  F is the Frobenius norm that is equivalent to the mixed (2, 2)-norm  · 2,2 . In this problem, we consider α = 1 and β = 2, which is a sum-of-norm constraint. Accordingly, the MMVTM LSτ problem is reformulated as (MMVTM LSτ ) min  J − Y  F s.t.  J 1,2 ≤ τ and the Pareto defined as

curve

for

the

MMV

φMMVTM (τ ) =  Jτ − Y  F

model

(20) is (21)

where Jτ is the optimal solution to the LSτ problem (20). According to [29, Th. 2.2] and [40, Ch. 5], φMMVTM (τ ) is continuously differentiable and  φMMV (τh ) = − TM

 H ( Jτh − Y )∞,2  Jτh − Y  F

(22)

where  · ∞,2 is the dual norm of  · 1,2 [29, Corollary 6.2]. σ Similarly, the root of the nonlinear equation φMMVTM (τ ) =

can also be reached by Newton iterations τh+1 = τh +

σ − φMMVTM (τh ) .  φMMV (τh ) TM

(23)

The projection operator Pτ [·] is replaced by an orthogonal projection onto  · 1,2 balls, Pτ,MMVTM [·], which is defined as follows:  Pτ,MMVTM [ J ] := arg min  J − S F s.t. S1,2 ≤ τ . (24) S

We refer to [29, Th. 6.3] for the implementation of the projection operator. The (MMVTM BP

σ ) problem is solved by

SUN et al.: LINEAR MODEL FOR MICROWAVE IMAGING OF HIGHLY CONDUCTIVE SCATTERERS

Algorithm 1 SPG for (MMVTM LSτ ) Problem Input : , Y , J , τ Output: Jτ , Rτ 1 Set minimum and maximum step lengths 0 < αmin < αmax ; 2 Set initial step length α0 ∈ [αmin , αmax ] and sufficient descent parameter γ ∈ (0, 0.5); 3 Set an integer line search history length M ≥ 1; H 4 J0 = Pτ,MMVTM [ J ], R0 = Y −  J0 , G 0 = − R0 ,  = 0; 5 begin      Tr Y H R −τ G  ∞,2  

compute 6 δ ← R  F −   R  F duality gap; 7 If δ ≤ , then break; 8 α ← α

initial step length; 9 begin 10 J ← Pτ,MMVTM [ J − αG  ]

projection; 11 R ← Y −  J update the corresponding residual;  2 R−h 2F + 12 if  R F ≤ max h∈[0,min{,M−1}]    H γ  Tr J − J G p, then 13 break; 14 else 15 α ← α/2; 16 end 17 end 18 J+1 ← J , R+1 ← R, G +1 ← − H R+1 update iterates; 19 J ← G ← G +1 − G  ;  J+1 − J ,  20 if  Tr  J H G ≤ 0 then 21 α+1 ← αmax update the Barzilai-Borwein step length; 22 else    

23 24 25 26 27

Tr  J H  J



α+1 ← min αmax , max αmin , {Tr{ J H G }}

;

end  ←  + 1; end return Jτ ← J , Rτ ← R ;

Algorithms 1 and 2 with the Pareto curve, φMMVTM (τ ), its  (τ ), and the projecderivative with respective to τ , φMMV TM tion operator, Pτ,MMVTM [·], defined by (21), (22), and (24), respectively.

E. Solving the MMV Model: TE Case For the TE polarization case, the electric field is not a scalar anymore. Therefore, care must be given to the formulation of the (MMVTEBP

σ ) problem. Considering the two components of electric field, E x and E y , the inverse scattering problem for the TE case can be formulated as (MMVTE BP

σ (25) σ ) min κTE ( J ) s.t. ρ( J − Y ) ≤

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Algorithm 2 Newton Root-Finding Framework Input : , Y ,

σ Output: J

σ 1 J0 ← 0, R0 ← Y , τ0 ← 0, h ← 0; 2 begin 3 If |Rh  F −

σ | ≤ , then break; 4 Solve the (MMVTM LSτ ) problem for τh using Algorithm 1; 5 Rh ←  Jh − Y ;

σ −φ (τ ) 6 τh+1 ← τh + φ  MMVTM(τ )h

Newton update; 7 8 9

MMVTM

h ← h + 1; end return J

σ ← Jh ;

h

where κTE ( J ) :=

N 

[ J2n−1,: J2n,: ]T 2

(26)

n=1

and ρ(·) :=  ·  F

(27)

are gauge functions [41]. The MMVTE LSτ problem is formulated accordingly as (MMVTE LSτ ) min ρ( J − Y ) s.t. κTE ( J ) ≤ τ. (28) 1) Derivation of the Dual: Let us rewrite (28) in terms of J and an explicit residual term R min ρ(R) s.t.  J + R = Y , κTE ( J ) ≤ τ. J ,R

(29)

The dual to this equivalent problem is given by [42, Ch. 5] max G(Z, λ) s.t. λ ≥ 0 Z,λ

(30)

where Z ∈ C(2M)×P and λ ∈ C are dual variables, and G is the Lagrange dual function, given by G(Z, λ) := inf {ρ(R) − Tr{Z H ( J + R − Y )} J ,R

+ λ(κTE ( J ) − τ )} (31)

where Tr represents the trace of a matrix. By separability of the infimum over J and R, we can rewrite G in terms of two separate suprema G(Z, λ) = Tr{Y H Z} − τ λ − sup{Tr{Z H R} − ρ(R)} R

− sup{Tr{Z H ( J )} − λκTE ( J )}. (32) J

It is easy to see that the first supremum is the conjugate function of ρ and the second supremum is the conjugate function of κTE [42, Ch. 3.3], by noting that Tr{Z H R} = vec{Z} H vec{R} ρ(R) = ρ(vec{R})

(33)

and Tr{Z H ( J )} = vec{

Z} H vec{ J } κTE ( J ) = κTE (vec{ J })

(34)

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 3, MARCH 2018

respectively. Here, vec{·} is the vectorization of a matrix,

Z=  H Z ∈ C(2N)×P , and κTE (vec{ J }) is defined equivalently as κTE ( J ) in (26). Therefore, we have  0, ρ o (Z) ≤ 1 H Tr{Z R} − ρ(R) = (35) ∞, otherwise and

 Tr{Z ( J )} − λκTE ( J ) = H

o (

Z) ≤ λ 0, κTE ∞, otherwise

(36)

With a simple matrix transformation of J n,: = [ J2n−1,: J2n,: ] and X n,: = [X 2n−1,: X 2n,: ], we can rewrite (44) as follows:   arg min  J − X F s.t. X1,2 ≤ τ

= Pτ,MMVTM [ J]. (45)

X

In doing so, the projection operator in TE case satisfies [29, Th. 6.3]. The (MMVTE BP

σ ) problem is solved by Algorithms 1 and 2 with the Pareto curve, its derivative with respect to τ , and the projection operator replaced by  (τ ), and Pτ,MMVTE [·], respectively. φMMVTE (τ ), φMMV TE

where the polar of ρ and the polar of κTE are defined by ρ o (Z) := sup {Tr{Z H R}|ρ(R) ≤ 1}

(37)

R

and o

( Z) := sup {Tr{Z H ( J )}|κTE( J ) ≤ λ} κTE

(38)

J

respectively. If the gauge function is a norm, the polar reduces to the dual norm [42, Sec. 3.3.1], i.e., ρ o (Z) = Z F and

N 1/∞  ∞ o

[

Z2n−1,:

κ ( Z) = Z2n,: ] TE

2

n=1

   = max [

Z2n−1,:

(39) Z2n,: ]2 |n = 1, 2, . . . , N (for more details, see [29, Corollary 6.2]). Substitution of (35) and (36) into (32) yields o

( Z) ≤ λ. (40) max Tr{Y H Z} − τ λ s.t. ρ o (Z) ≤ 1, κTE Z,λ

In the case ρ(·) =  ·  F , the dual variable Z can be easily derived from sup Tr{Z H R} − R F = 0, if Z F ≤ 1

(41)

R

which is Z = (R/R F ). To derive the optimal λ, we can observe from (40) that as long as τ > 0, λ must be at its o

( Z); otherwise, one can increase the objective lower bound κTE H Tr{Y Z} − τ λ. Therefore, we obtain λ=

(42)

According to [40, Th. 5.2], we know that, on the open interval τ ∈ (0, τ0 ), where τ0 = min {τ ≥ 0|φMMVTE (τ ) = min ρ(R)} J

the Pareto curve φMMVTE (τ ) = ρ(R) is strictly decreasing, and continuously differentiable with o ( H R) κTE . R F

(43)

The projection operator Pτ,MMVTM [·] is replaced by an orthogonal projection onto κTE (·) balls, Pτ,MMVTE [·], which is defined as follows:  Pτ,MMVTE [ J ] := arg min  J − S F s.t. κTE (S) ≤ τ . (44) S

In real applications, the noise level, i.e., the parameter

σ, is generally unknown, which means the termination condition, σ , does not work anymore. In order to deal with φMMV (τ ) =

this problem, we modified the SPGL1 method based on the CV technique [30], [31], in which

σ is set zero and the iteration is terminated using CV technique. In doing so, the problem of estimating the noise level, i.e., the parameter

σ , can be well circumvented. CV is a statistical technique that separates a data set into a training (estimation) set and a testing (CV) set. The training set is used to construct the model and the testing set is used to adjust the model order so that the noise is not overfitted. The basic idea behind this technique is to sacrifice a small number of measurements in exchange of prior knowledge. Specifically, when CV is utilized in the SPGL1 method, we separate the original scattering matrix to a reconstruction matrix  p,r ∈ C Q r ×N and a CV matrix  p,C V ∈ C Q CV ×N with Q = Q r + Q C V . The measurement vector y p is also separated accordingly, to a reconstruction measurement vector y p,r ∈ C Q r and a CV measurement vector y p,C V ∈ C Q CV . The reconstruction residual and the CV residual are defined as ⎛ ⎞1/2 P   y p,r −  p,r j p 22 ⎠ (46) rrec = ⎝ p=1

and

o ( H R) κTE . R F

 φMMV (τ ) = −λ = − TE

F. CV-Based Modified SPGL1

⎛ rCV = ⎝

P 

⎞1/2  y p,C V −  p,C V j p 22 ⎠

(47)

p=1

respectively. In doing so, every iteration can be viewed as two separate parts: reconstructing the contrast sources by SPGL1 and evaluating the outcome by the CV technique. The trend of CV residual in iteration behaves abruptly different (turns from decreasing to increasing) comparing with that of reconstruction residual, as soon as the reconstructed signal starts to overfit the noise. The reconstructed contrast sources are selected as the output on the criterion that its CV residual is the smallest one. In order to find the smallest CV residual, a maximum number, Nmax , is needed and set a large value to guarantee the smallest CV residual occurs in the range of the Nmax iterations. In this case, a large number of iterations are performed in vain,

SUN et al.: LINEAR MODEL FOR MICROWAVE IMAGING OF HIGHLY CONDUCTIVE SCATTERERS

which decreases the efficiency of the algorithm. Therefore, we consider an alternative termination condition given by NIter > Nopt + N

γMMVTM [n] =

P 

| j p,n |2

x j -axis, which are given by 1 (1) ωμ0 H0 (−k R) 4

(1)  H1 (−k R) kx 22 (1) −k + 2 H2 (−k R) = 4ωε0 R R

E 3,3 =

(53a)

E 1,1

(53b)

(48)

where NIter is the current iteration number, Nopt is the iteration index corresponding to the smallest CV residual—the optimal solution. The idea behind this criterion is that the CV residual is identified as the smallest one if the CV residual keeps monotonously increasing for N times of iteration. In the following experimental examples, this termination condition works well with N = 30. Once the normalized contrast sources are obtained, one can achieve the shape of the scatterers defined as

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k 2 x 1 x 2 (1) H (−k R) 4ωε0 R 2 2 k 2 x 1 x 2 (1) = H (−k R) 4ωε0 R 2 2

(1)  H1 (−k R) kx 12 (1) −k + 2 H2 (−k R) = 4ωε0 R R

E 1,2 =

(53c)

E 2,1

(53d)

E 2,2

where R = xs − xr 2 . Equations (51) and (52) can be reformulated as a set of linear systems of equations

(49)

Y g xs = f xs

p=1

or γMMVTE [n] =

P 

(| j p,2n−1 |2 + | j p,2n |2 )

(50)

p=1

with n = 1, 2, . . . , N, where j p,n , γMMVTM [n], and γMMVTE [n] represent the nth element of vector j p , γMMVTM , and γMMVTE , respectively. In the end of this section, we remark that as the regularized solution corresponds to the least sum of norm, the nonmeasurable equivalent contrast sources [43] tend to be ignored.

In this section, the proposed method is tested with both synthetic data and experimental data. In the meanwhile, we have also processed the same data using LSM for comparison. Since the background of the experiments is free space, the LSM method consists in solving the integral equation of the indicator function  (51) E 3 (xr , xt )g3 (xs , xt )d xt = E 3,3 (xs , xr ) and  

E1 0

  0 g (xr , xt ) 1,1 g2,1 E2

 g1,2 (xs , xt )d xt g2,2   E 1,1 E 1,2 (xs , xr ) (52) = E 2,1 E 2,2

for the TM and TE cases, respectively, where E 1 (xr , xt ), E 2 (xr , xt ), and E 3 (xr , xt ) represent x 1 , x 2 , and x 3 components of the scattered field probed at xr corresponding to the transmitter at xt , respectively; xs is the sampling point in the inversion domain; and E i, j (xs , xr ) is i th component of the electric field at xr generated by an ideal electric dipole located at xs with the polarization vector parallel to the

(54)

where Y is the measurement data matrix, g xs is the indicator function of the sampling point xs in the form of a column vector, and f xs is the right side of (51) in the form of a column vector. Following the same approach in [44] and [45] for solving (54), the shape of the scatterers is defined by: γLSM (xs ) = where g xs 2 is given by g xs 2 =

D  d=1

III. L INEAR S AMPLING M ETHOD AND I TS I MPROVED V ERSION

(53e)

1 g xs 2

sd sd2 + a 2

2

 H 2 u f x  s d

(55)

(56)

where sd represents the singular value of matrix Y corresponding to the singular vector ud , D = min{P, Q}, and a = 0.01 × maxd {sd }. We have also considered, in the TM cases, the improved LSM proposed in [12] in the comparison of the proposed method and LSM. The indicator function of improved LSM is defined as ⎛ ⎞1   2I y I  x 2  gi,xs gi,xs 2 ⎠ , I = ka (57) γLSM,I (xs ) = ⎝  2  g xs  g xs 2 i=1 where a is the radius of a smallest ball that covers the x targets, the power (1/2I ) is the normalization factor, and gi,x s y and gi,xs are obtained by replacing E 3,3 (xs , xr ) in (51) with y ϕix (xs , xr ) and ϕi (xs , xr ), respectively 1 ωμ0 Hi(1)(−k R) cos(i (φr − φs )) (58a) 4 1 y (58b) ϕi = ωμ0 Hi(1)(−k R) sin(i (φr − φs )) 4 where φr and φs are the angular components of xs and xr in polar coordinate system, respectively. We refer to [12] for more details of this indicator function. It is worth mentioning that both the contrast source j p and the indicator function g x are proportional to the amplitude of the electric field. According to the definition in (49), (50), and (55), γMMV and γLSM are proportional and inversely ϕix =

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Fig. 3.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 3, MARCH 2018

Measurement configuration of Simulations 1 and 2.

proportional to the power of the electric field, respectively. Therefore, the decibel scaling shown in the following examples is defined as: ! γ . (59) γdB = 10 × log10 max{γ } IV. S YNTHETIC DATA I MAGING In this section, the proposed method is tested with synthetic data. The transmitting antenna is simulated for simplicity with an ideal electric dipole (TM-polarization case) and an ideal magnetic dipole (TE-polarization case). Coordinate system is established such that the dielectric parameters are variable along the x and y axes, but invariable along the z-axis. The transmitting antenna rotates on a circular orbit of 3 m radius centering at the origin (0, 0). The receiving positions are taken on the same orbit without any position close than 30° from the transmitting antenna. The measurement configuration of Simulations 1 and 2 is shown in Fig. 3, in which the selection of CV measurements and reconstruction measurements is illustrated. Empirically, an arc length ≥ λ/3 is a good selection. The number of the CV receivers on each arc depends on how dense the receiver positions are, and the total CV receiver number is around 20% of the total measurement number [31]. The operating frequency is 500 MHz. Two configurations of different objects are considered. One is combined with two circular metallic cylinders and the other one is a metallic cylinder with a “crescent-shaped” cross section. The radius of the circular cross section is 0.2 m (= λ/3), and the centers of the two circles are (−0.45, 0.6) and (0.45, 0.6), respectively. The crescent is the subtraction of two circles of radius 0.6 m (= λ) centering at (0, 0) and (0.4, 0) [see Figs. 6(a) and 10(a) for their true geometry]. The forward EM scattering problem is solved by a MATLABbased 3-D FDFD package “MaxwellFDFD” [32]. The technique of nonuniform staggered grids is used to reduce the computational burden, while for inverting the measurement data, we consider uniform discretization such that an inverse crime is circumvented. In the forward solver, we consider √ a fine grid size of λ/(45 r ). The data for inversion are

Fig. 4. Correlation coefficient curves in terms of transmitter number in Simulations 1 and 2. Receiver number is fixed to 151; 10- and 30-dB Gaussian random additive noises are considered, respectively. (a) TM-polarized data. (b) TE-polarized data.

obtained by subtracting the incident field from the total field. Periodic boundary conditions are imposed on the design of the FDFD stiffness matrix in order to simulate the 2-D configuration. Perfect matching layer is used to simulate an anechoic chamber environment. A. Determine the Measurement Configuration To determine the measurement configuration, we need to investigate the relationship between the transceiver numbers and the imaging quality. Let us first consider 120 transmitters and 151 receivers, i.e., the transmitter rotates on the circular orbit with a step of 3°, and the receiver rotates on the measurement arc of 300° with a step of 2°. The CV receivers are selected in the same way as shown in Fig. 3, but there are four continuous CV receiver positions in each CV arc (equivalent to 8°). Now let us disturb the measurement data

SUN et al.: LINEAR MODEL FOR MICROWAVE IMAGING OF HIGHLY CONDUCTIVE SCATTERERS

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Fig. 6. Scatterer geometry and its reconstructed shapes in Simulation 1; 30-dB Gaussian noise is added to the measurement data. (a) Scatterer geometry. The scatterer shape (the value of the indicator function in decibels) reconstructed by processing the TM-polarized data with (b) MMV, (c) LSM, and (d) improved LSM with I = 7, respectively.

Fig. 5. Correlation coefficient curves in terms of receiver number in Simulations 1 and 2. Transmitter number is fixed to 18 and 120, respectively; 10- and 30-dB Gaussian random additive noises are considered, respectively. (a) TM-polarized data. (b) TE-polarized data.

(the scattered fields) with Gaussian additive random noise of 30 dB signal-to-noise ratio (SNR), and then process the data by the proposed method. If we use the reconstructed image as the reference image, denoted by γref , then a correlation coefficient can be defined as "N (γref [n] − γref )(γ [n] − γ ) rcorr := #" n=1 (60) "N N 2 2 (γ [n] − γ ) (γ [n] − γ ) ref ref n=1 n=1 where γ denotes the MMV image with different measurement configurations and noise levels, and γref and γ are the mean values of γref and γ , respectively. The correlation coefficient reflects the similarity degree of two images. The minor negative correlation coefficients are forced to zeros, as negative correlation does not make any sense for two amplitude images. Now we first fix the receiver number to 151, and calculate the correlation coefficients of Simulations 1 and 2 with different transmitter numbers. Fig. 4(a) and (b) shows the correlation coefficient curves in terms of transmitter number

by processing the TM-polarized data and TE-polarized data, respectively. Two SNRs, 10 and 30 dB, are considered. From Fig. 4, we observe that an obvious decrease in correlation coefficient occurs at 18 transmitters, indicating that the image quality gets worse when the transmitter number is less than 18. The correlation coefficient curves of 10 and 30 dB maintain the same trend, and the correlation coefficients of 10 dB maintains above 0.95 when more than 18 transmitters are used, indicating the proposed method is robust against the Gaussian additive random noise. Then we fix the transmitter number to 18 and 120, respectively, and image the targets in Simulations 1 and 2 with different receiver numbers. Since CV technique needs enough amount of measurements, the noise level is assumed exactly known when the receiver number is less than or equal to 31. Fig. 5(a) and (b) shows the correlation coefficient curves in terms of receiver number by processing the TM-polarized data and TE-polarized data, respectively. Two SNRs, 10 and 30 dB, are considered. From Fig. 5, we observe that the smallest receiver number to ensure a stable imaging quality is 16. The correlation coefficient of 18 transmitters and 10 dB SNR maintains rcorr ≥ 0.90 when the receiver number ≥16. Since the reference image in the definition of the correlation coefficient is not the real shape of the targets, it is actually an

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Fig. 7. Scatterer geometry and its reconstructed shapes in Simulation 1. 10 dB Gaussian noise is added to the measurement data. The scatterer shape (the value of the indicator function in dB) reconstructed by processing the TM-polarized data with MMV (a), LSM (b), and improved LSM with I = 7 (c), respectively.

asymptotic measure of the imaging quality. Discussion of the imaging results is given in the next section in comparison with the LSM images to further investigate the imaging performance. In Sections IV-B and V, we select 18 transmitters (equivalent to an interval of 20°) for two reasons: 1) the proposed method works well in the numerical simulations with 18 transmitters and 2) more experiments and targets are required to demonstrate the good imaging performance when 18 transmitters are used. Note that the noise level is not available in real applications; 61 receivers (equivalent to an interval of 5°) are selected in the numerical experiments for the use of CV technique. B. Imaging Results 1) Simulation 1: To reduce the computational cost, we restrict the inversion domain to [−1.0, 1.0] × [−0.4, 1.6] m2 . The inversion domain is discretized with an equal grid size of 0.01 m (= λ/60). Let us first consider the TM-polarized data disturbed with Gaussian additive random noise of 30 dB SNR. The residual curves are shown in Fig. 8(a). The trend of the residual curves is like staircases, and each step corresponds to one update of the parameter τ . The CV residual starts to increase after 80 iterations, and N = 30 more iterations are

Fig. 8. Reconstruction residual and CV residual curves by processing the TM-polarized data of Simulation 1. (a) SNR = 30 dB. (b) SNR = 10 dB.

performed before termination. The solution of the minimum CV residual is selected as the optimal solution. The scatterer shape reconstructed by MMV, LSM, and improved LSM with I = 7 is shown in Fig. 6(b)–(d), respectively. By comparison of Fig. 6(c) and (d), it is observed that the artifacts between the two circular cylinders are suppressed by improved LSM. However, the average amplitude of the sidelobes in the region of no targets increases from −15 to −10 dB. From Fig. 6(b), we observe that the proposed method shows higher resolution and lower sidelobes in comparison with Fig. 6(c) and (d), indicating that the resolving ability of the proposed method is better than LSM. To study the imaging performance with different SNRs, let us decrease the SNR to 10 dB, and the images results are shown in Fig. 7. By comparing Figs. 6 and 7, we can observe obvious degradation in the LSM images, while there is no obvious degradation in the MMV images. Fig. 8(b) shows the residual curves, from which we can see the reconstruction residual of the optimal solution, rrec ≈ 0.105, is higher than that of Fig. 8(a), rrec ≈ 0.025. Now let us process the TE-polarized data of different SNRs, 30 and 10 dB. Fig. 9 shows the scatterer shape reconstructed by MMV and LSM, respectively. The imaging results demonstrate again that, in the perspective of resolving ability, the proposed method outperforms the LSM. In addition, the proposed method maintains good imaging performance for different SNRs.

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Fig. 9. Scatterer shape (the value of the indicator function in dB) reconstructed by processing the TE-polarized data of Simulation 1. (a), (c): MMV images; (b), (d): LSM images; (a), (b): SNR= 30 dB; (c), (d): SNR= 10 dB.

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Fig. 10. Scatterer geometry and its reconstructed shapes in Simulation 2; 30-dB Gaussian noise is added to the measurement data. (a) Scatterer geometry. The scatterer shape (the value of the indicator function in decibels) reconstructed by processing the TM-polarized data with (b) MMV, (c) LSM, and (d) improved LSM with I = 6, respectively.

C. Analysis of Computational Complexity 2) Simulation 2: In the second simulation, we restrict the inversion domain to [−1.0, 1.0]×[−1.0, 1.0] m2 , in which the target is fully covered. The inversion domain is discretized with an equal grid size of 0.01 m (= λ/60). First, we process the TM-polarized data of 30-SNR by MMV and LSM, respectively. Fig. 10(b)–(d) shows the reconstructed shape by MMV, LSM, and improved LSM, respectively. We can see from the results that the boundary at the left side is well reconstructed by the three methods, while the arch at the right side shows more artifacts in Fig. 10(b) and (c), because the arch at the right side is concave and multipath scattering is more severe than the left side that is convex. Comparison of Fig. 10(c) and (d) shows minor suppression of artifacts in the interior of the cylinder by improved LSM. The imaging results of 10-dB SNR data are shown in Fig. 11. Apart from some minor artifacts, no obvious degradation occurs in MMV image, while we can observe severe degradation of image resolution in LSM images. The MMV image and LSM image obtained by processing the TE-polarized data of 30 and 10 dB SNR are shown in Fig. 12. From the results, we can observe that the proposed method is able to reconstruct the scatterer’s shape with some artifacts occurred at the concave side, while LSM fails to give the basic profile of the target. Considering the length of this paper, the residual curves in this simulation are not given.

The sensing matrix  can be computed (or analytically given for the experiments in homogeneous background) and stored beforehand. It is easy to see from Algorithms 1 and 2 that the computational complexity of the GMMV-based linear method primarily depends on the number of matrix–vector multiplications,  J , and  H R. Empirically, each iteration involves maximally two times of  J and one time of  H R. In order to study the computational complexity of the proposed algorithms, we use one complex data multiplication as the measurement unit. The computational complexity for computing  J and  H R is Q N for the TM case and 4Q N for the TE case. Let us use L to denote the iteration number, then the computational complexity of the proposed method in the TM and TE case is CTM = 3L Q N

(61a)

CTE = 12L Q N

(61b)

respectively. When the mesh gets finer or the inversion domain gets larger, the iteration number, L, almost keeps unchanged, and the running time therefore linearly increases with the grid number. In our experiments, the imaging algorithms are implemented with MATLAB language. We ran the program on a desktop with one Intel Core i5-3470 CPU @ 3.20 GHz, and we did not use parallel computing. Table I lists the running times of

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Fig. 11. Scatterer geometry and its reconstructed shapes in Simulation 2. 10 dB Gaussian noise is added to the measurement data. The scatterer shape (the value of the indicator function in dB) reconstructed by processing the TM-polarized data with MMV (a), LSM (b), and improved LSM with I = 7 (c), respectively.

the proposed method, LSM, and improved LSM in the two simulations. As one can see that the computation time of the proposed method is hundreds of times longer than that of LSM and tens of times longer than improved LSM. The most timeconsuming part of the proposed method is the matrix–vector multiplication in each iteration, while LSM calls only singular value decomposition to the measurement data matrix for once. However, the running times of the proposed method are still acceptable in view of the gain of resolving ability. V. E XPERIMENTAL DATA I MAGING In this section, we applied our method to the experimental data provided by the Remote Sensing and Microwave Experiments Team at the Institut Fresnel, France, using an HP8530 network analyzer [33]. The experimental setup consists of a large anechoic chamber, 14.50 m long, 6.50 m wide, and 6.50 m high, with a set of three positioners to adjust antennas or target positions. A 2-D bistatic measurement system is considered, with an emitter placed at a fixed position on the circular rail, while a receiver is rotating with the arm around a vertical cylindrical target. The targets rotated from 0° to 350° in steps of 10° with a radius of 720 ± 3 mm, and the receiver rotated from 60° to 300° in steps of 5° with a radius of 760 ± 3 mm. In TE polarization, only

Fig. 12. Scatterer shape (the value of the indicator function in dB) reconstructed by processing the TE-polarized data of Simulation 2. (a), (c): MMV images; (b), (d): LSM images; (a), (b): SNR= 30 dB; (c), (d): SNR= 10 dB.

the component orthogonal to both the invariance axis of the cylinder and the direction of illumination is measured. Fig. 13 gives the measurement configuration, in which the selection of CV measurements and reconstruction measurements is illustrated. The time dependence in this experiment is exp(i ωt). Therefore, after subtracting the incident field from the total field, the measurement data can be directly used for inversion. The targets we consider here are a rectangular metallic cylinder and a “U-shaped” metallic cylinder, which have been shown in Fig. 14(a) and (b). Three data sets are processed: rectTM_cent at 16 GHz, uTM_shaped at 8 GHz, and rectTE_8f at 16 GHz. The selected frequencies correspond to wavelengths that are comparable with the dimension of the targets, i.e., the resonance frequency range. As we have discussed in Section IV-A, 18 transmitters of 20° arc interval are enough for the proposed method to give good resolution. Therefore, in the following experiments, only 18 transmitter positions (half of the measurement data) are used for imaging. First, let us process the TM-polarized data set: rectTM_cent at 16 GHz. The inversion domain for imaging the rectangular metallic cylinder is restricted to [−50, 50] × [−50, 50] mm2 , and the inversion domain is discretized with equal grid size of 0.5 mm (= λ/37.5). Fig. 15(a)–(c) shows the scatterer shape reconstructed by MMV, LSM, and improved LSM, respectively. The residual curves are shown in Fig. 18(a).

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TABLE I RUNNING T IMES OF THE T WO N UMERICAL E XAMPLES

Fig. 13. Measurement configuration of the Fresnel data sets: rectTM_cent, uTM_shaped, and rectTE_8f.

Fig. 15. Scatterer shape (the value of the indicator function in decibels) reconstructed by processing the TM-polarized data set: rectTM_cent at 16 GHz with (a) MMV, (b) LSM, and (c) improved LSM with I = 9, respectively. A total of 18 transmitter positions and 49 receiver positions for each transmitter are selected for imaging.

TABLE II RUNNING T IMES OF THE E XPERIMENTAL E XAMPLES

Fig. 14. Geometry of the scatterers. (a) Rectangular metallic cylinder. (b) “U-shaped” metallic cylinder.

From the imaging results, we observe that Fig. 15(a) shows higher resolution and less artifacts than Fig. 15(b) and (c). And we also observe that there is no big difference between Fig. 15(b) and (c). Now let us consider the “U-shaped” metallic cylinder. The inversion domain is restricted to [−100, 100] × [−100, 100] mm2 , and the inversion domain is discretized with an equal grid size of 1 mm (= λ/37.5). Fig. 16(a)–(c) gives the scatterer shape reconstructed by MMV, LSM, and improved LSM, respectively. The residual curves are shown in Fig. 18(b). Severe artifacts can be observed in LSM image and improved LSM image. Furthermore, the suppression to the artifacts is not obvious in the improved LSM image. In the contrary, the “U-shaped” cylinder is reconstructed by

the proposed method with the boundary well distinguished. Some artifacts can be observed vertically aligned in the interior and below the opening, which are caused by the complicated scattering in the opening area. Finally, let us process the TE-polarized data set: rectTE_8f at 16 GHz. The scatterer shape reconstructed by MMV and LSM is shown in Fig. 17(a) and (b), respectively. The residual curves are shown in Fig. 18(c). It can be observed that the boundary of the rectangular metallic cylinder is not distinguishable in Fig. 17(b), while in Fig. 17(a), the rectangular boundary can be clearly distinguished. The data processing is performed on the same computing platform, and the running times of all the methods are listed in Table II. In the end, we summarize this section as follows: 1) the proposed method is

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Fig. 16. Scatterer shape (the value of the indicator function in decibels) reconstructed by processing the TM-polarized data set: uTM_shaped at 8 GHz with (a) MMV, (b) LSM, and (c) improved LSM with I = 8, respectively. A total of 18 transmitter positions and 49 receiver positions for each transmitter are selected for imaging.

Fig. 18. Reconstruction residual curve and CV residual curve of the Fresnel data sets. (a) rectTM_cent at 16 GHz. (b) uTM_shaped at 8 GHz. (c) rectTE_8f at 16 GHz.

VI. C ONCLUSION Fig. 17. Scatterer shape (the value of the indicator function in decibels) reconstructed by processing the TE-polarized data set: rectTE_8f at 16 GHz with (a) MMV and (b) LSM. A total of 18 transmitter positions and 49 receiver positions for each transmitter are selected for imaging.

able to obtain higher resolution than LSM and improved LSM; 2) the proposed method is more computationally expensive than LSM and improved LSM; and 3) the suppression to the artifacts by improved LSM is not obvious when less transmitters are used.

In this paper, we addressed the nonlinear inverse scattering problem of highly conductive objects with a linear model. MMV model is exploited and sum-of-norm constraint is used as a regularization approach. A CV-based modified SPGL1 method is proposed to invert the data without estimating the noise level. Numerical results and experimental results of both TM polarization and TE polarization demonstrate that the proposed method shows higher resolving ability than both LSM and improved LSM. In some cases where the latter two methods fail, the proposed method can still successfully recover the profile of the targets. Numerical experiments also

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demonstrate that the MMV method maintains good imaging performance in the disturbance of 10-dB SNR Gaussian random noise. The running time of the proposed method is hundreds of times longer than LSM and tens of times longer than improved LSM. However, it is still promising in view of the gain of resolving performance and the linear relationship between the computational complexity and the size of the inversion domain. In addition, it might fail in the presence of not conductive scatterers, which is an obvious limitation of the proposed method. R EFERENCES [1] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Mineola, NY, USA: Dover, 2014. [2] R. Kleinman and P. M. Van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci., vol. 29, no. 4, pp. 1157–1169, 1994. [3] P. M. Van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse problems, vol. 13, no. 6, pp. 1607–1620, 1997. [4] O. Féron, B. Duchêne, and A. Mohammad-Djafari, “Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data,” Inverse Problems, vol. 21, no. 6, pp. S95–S115, 2005. [5] C. Yu, L.-P. Song, and Q. H. Liu, “Inversion of multi-frequency experimental data for imaging complex objects by a DTA–CSI method,” Inverse Problems, vol. 21, no. 6, pp. S165–S178, 2005. [6] S. Sun, B. J. Kooij, T. Jin, and A. G. Yarovoy, “Cross-correlated contrast source inversion,” IEEE Trans. Antennas Propag., vol. 65, no. 5, pp. 2592–2603, May 2017. [7] F. Di Benedetto, C. Estatico, J. G. Nagy, and M. Pastorino, “Numerical linear algebra for nonlinear microwave imaging,” Electron. Trans. Numer. Anal., vol. 33, pp. 105–125, Oct. 2009. [8] A. J. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1600–1610, May 2005. [9] O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imag., vol. 30, no. 5, pp. 1129–1142, May 2011. [10] D. Colton and A. Kirsch, “A simple method for solving inverse scattering problems in the resonance region,” Inverse problems, vol. 12, no. 4, pp. 383–393, 1996. [11] D. Colton, M. Piana, and R. Potthast, “A simple method using Morozov’s discrepancy principle for solving inverse scattering problems,” Inverse Problems, vol. 13, no. 6, pp. 1477–1493, 1997. [12] L. Crocco, L. Di Donato, I. Catapano, and T. Isernia, “An improved simple method for imaging the shape of complex targets,” IEEE Trans. Antennas Propag., vol. 61, no. 2, pp. 843–851, Feb. 2013. [13] S. N. Fata and B. B. Guzina, “A linear sampling method for near-field inverse problems in elastodynamics,” Inverse Problems, vol. 20, no. 3, pp. 713–736, 2004. [14] T. Arens, “Why linear sampling works,” Inverse Problems, vol. 20, no. 1, pp. 163–173, 2003. [15] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93, 3rd ed. New York, NY, USA: Springer, 2013. [16] M. R. Eskandari, R. Safian, and M. Dehmollaian, “Three-dimensional near-field microwave imaging using hybrid linear sampling and level set methods in a medium with compact support,” IEEE Trans. Antennas Propag., vol. 62, no. 10, pp. 5117–5125, Oct. 2014. [17] D. J. Daniels, Ground Penetrating Radar. Hoboken, NJ, USA: Wiley, 2005. [18] A. Roger, “Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem,” IEEE Trans. Antennas Propag., vol. AP-29, no. 2, pp. 232–238, 1981, doi: 10.1109/TAP.1981.1142588. [19] A. Qing, “Electromagnetic inverse scattering of multiple twodimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1251–1262, Jun. 2003. [20] A. Qing, “Electromagnetic inverse scattering of multiple perfectly conducting cylinders by differential evolution strategy with individuals in groups (GDES),” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1223–1229, May 2004.

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[21] S. Caorsi, A. Massa, and M. Pastorino, “A crack identification microwave procedure based on a genetic algorithm for nondestructive testing,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 1812–1820, Dec. 2001. [22] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems, vol. 25, no. 12, pp. 123003-1–123003-41, 2009. [23] M. Salucci, L. Poli, N. Anselmi, and A. Massa, “Multifrequency particle swarm optimization for enhanced multiresolution GPR microwave imaging,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 3, pp. 1305–1317, 2017. [24] L. Poli, G. Oliveri, F. Viani, and A. Massa, “MT–BCS-based microwave imaging approach through minimum-norm current expansion,” IEEE Trans. Antennas Propag., vol. 61, no. 9, pp. 4722–4732, Sep. 2013. [25] S. Sun, B. J. Kooij, and A. G. Yarovoy, “Solving the PEC inverse scattering problem with a linear model,” in Proc. URSI Int. Symp. Electromagn. Theory (EMTS), Aug. 2016, pp. 144–147. [26] E. Van Den Berg and M. P. Friedlander, “Theoretical and empirical results for recovery from multiple measurements,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2516–2527, May 2010. [27] S. Sun, B. J. Kooij, and A. Yarovoy, “Linearized three-dimensional electromagnetic contrast source inversion and its applications to halfspace configurations,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 6, pp. 3475–3487, Jun. 2017. [28] E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput., vol. 31, no. 2, pp. 890–912, 2008. [29] E. Van den Berg and M. P. Friedlander, “Sparse optimization with leastsquares constraints,” SIAM J. Optim., vol. 21, no. 4, pp. 1201–1229, 2011. [30] R. Ward, “Compressed sensing with cross validation,” IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5773–5782, Dec. 2009. [31] J. Zhang, L. Chen, P. T. Boufounos, and Y. Gu. (2016). “Cross validation in compressive sensing and its application of OMP-CV algorithm.” [Online]. Available: https://arxiv.org/abs/1602.06373 [32] W. Shin, “3D finite-difference frequency-domain method for plasmonics and nanophotonics,” Ph.D. dissertation, Dept. Elect. Eng., Stanford Univ., Stanford, CA, USA, 2013. [33] K. Belkebir and M. Saillard, “Special section: Testing inversion algorithms against experimental data,” Inverse Problems, vol. 17, no. 6, pp. 1565–1571, 2001. [34] S. R. Rengarajan and Y. Rahmat-Samii, “The field equivalence principle: Illustration of the establishment of the non-intuitive null fields,” IEEE Antennas Propag. Mag., vol. 42, no. 4, pp. 122–128, Aug. 2000. [35] S. Sun, G. Zhu, and T. Jin, “Novel methods to accelerate CS radar imaging by NUFFT,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 1, pp. 557–566, Jan. 2015. [36] R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. Roy. Stat. Soc. Ser. B, Methodol., vol. 58, no. 1, pp. 267–288, 1996. [37] E. G. Birgin, J. M. Martínez, and M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim., vol. 10, no. 4, pp. 1196–1211, 2000. [38] E. G. Birgin, J. M. Martínez, and M. Raydan, “Inexact spectral projected gradient methods on convex sets,” IMA J. Numer. Anal., vol. 23, no. 4, pp. 539–559, 2003. [39] Y.-H. Dai and R. Fletcher, “Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming,” Numer. Math., vol. 100, no. 1, pp. 21–47, 2005. [40] E. Van den Berg, “Convex optimization for generalized sparse recovery,” Ph.D. dissertation, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, Canada, 2009. [41] R. T. Rockafellar, Convex Analysis (Princeton Mathematics Series), vol. 28. Princeton, NJ, USA: Princeton Univ. Press, 1970. [42] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY, USA: Cambridge Univ. Press, 2004. [43] S. Caorsi and G. L. Gragnani, “Inverse-scattering method for dielectric objects based on the reconstruction of the nonmeasurable equivalent current density,” Radio Sci., vol. 34, no. 1, pp. 1–8, 1999. [44] I. Catapano, L. Crocco, and T. Isernia, “On simple methods for shape reconstruction of unknown scatterers,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1431–1436, May 2007. [45] L. Crocco, I. Catapano, L. Di Donato, and T. Isernia, “The linear sampling method as a way to quantitative inverse scattering,” IEEE Trans. Antennas Propag., vol. 60, no. 4, pp. 1844–1853, Apr. 2012.

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Shilong Sun received the B.S. and M.S. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2011 and 2013, respectively. He is currently pursuing the Ph.D. degree at the Microwave Sensing, Signals and Systems Group, Delft University of Technology, Delft, The Netherlands. His current research interests include inverse scattering problems and radar imaging.

Bert Jan Kooij was born in Amersfoort, The Netherlands, in 1959. He received the B.Sc. and M.Sc. degrees in electrical engineering and Ph.D. degree in technical sciences from the Delft University of Technology, Delft, The Netherlands, in 1984, 1986, and 1994, respectively. Since 1987, he has been a Scientific Staff Member with the Electromagnetic Research Group, Delft University of Technology, where he has carried out research and taught classes in the area of electromagnetics, as well as acoustics, wave propagation, and scattering problems. During a three-month period in 1996, he was a Visiting Scientist with the Ecole Supérieure dâ Electricité, Gif-sur-Yvette, France. He was involved in transient wave propagation problems in the field of elastodynamics and electromagnetics. Since 2010, he has been a member of the Microwave Sensing, Signals and Systems Group, Delft University of Technology. His current research interests include the computation of inverse wave-field problems employing iterative techniques based on error minimization and space time-domain wave-field modeling.

Alexander G. Yarovoy (F’15) received the Diploma degree (Hons.) in radiophysics and electronics, Candidate Phys. and Math. Sci. degree in radiophysics, and Doctor Phys. and Math. Sci. degree in radiophysics from Kharkov State University, Kharkov, Ukraine, in 1984, 1987, and 1994, respectively. In 1987, he joined the Department of Radiophysics, Kharkov State University, as a Researcher, where he became a Professor in 1997. From 1994 to 1996, he was with the Technical University of Ilmenau, Ilmenau, Germany, as a Visiting Researcher. Since 1999, he has been with the Delft University of Technology, Delft, The Netherlands. Since 2009, he has been the Chair of Microwave Sensing, Signals, and Systems. He has authored or co-authored over 250 scientific or technical papers and 14 book chapters, and holds 4 patents. His current research interests include ultrawideband microwave technology and its applications (particularly radars) and applied electromagnetics (particularly UWB antennas). Prof. Yarovoy was a co-recipient of the European Microwave Week Radar Award for the paper that best advances the state of the art in radar technology in 2001 (together with L. P. Ligthart and P. van Genderen) and in 2012 (together with T. Savelyev) and was also a co-recipient of the Best Paper Award of the Applied Computational Electromagnetic Society in 2010 (together with D. Caratelli). He has served as a Guest Editor of five Special Issues for IEEE T RANSACTIONS and other journals. Since 2008, he has been the Director of the European Microwave Association. Since 2011, he has been an Associate Editor of the International Journal of Microwave and Wireless Technologies. He has served as the Chair and TPC Chair of the fifth European Radar Conference, Amsterdam, The Netherlands, as well as the Secretary of the first European Radar Conference, Amsterdam. He has also served as the Co-Chair and TPC Chair of the 10th International Conference on Ground Penetrating Radar, Delft.

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Temporal Coupled-Mode Theory of Electromagnetic Components Described by Magnetic Groups of Symmetry Victor Dmitriev, Gianni Portela , and Leno Martins

Abstract— We consider peculiarities of application of the temporal coupled-mode theory to the electromagnetic components with magnetic symmetry. The coupled-mode theory is widely used for analysis of reciprocal devices described by symmetrical scattering matrices and by unitary elements of symmetry, such as for example, a plane of symmetry. However, nonreciprocity and low symmetry of components described by antiunitary elements lead to a necessity of modification of the method. Using as an example a photonic crystal-based W-circulator described by an antiplane of symmetry, we show applicability of the method to the components with antiunitary elements. The circulator characteristics are calculated and compared with those obtained by a numerical method, demonstrating good agreement between the two methods. Index Terms— Circulators, magnetic group theory, photonic crystals (PhCs), temporal coupled-mode theory (TCMT).

I. I NTRODUCTION ANY nonreciprocal and control components based on magnetooptical (MO) materials in millimeter wave, terahertz, and optical regions have been suggested during the last 15 years. Among them are circulators of different types [1]–[3], switches [4], dividers [5], and multifunctional devices [6]. Circulators protecting sources of electromagnetic waves from undesirable reflections in circuits at microwaves are fulfilled in the metal waveguide and the microstrip technology [7]. In millimeter wave, terahertz, and optical regions, circulators can be implemented by using 2-D photonic crystals (PhCs) with ferrites or magnetoplasma materials [1], [3], [8]. Analysis of such structures is usually based on numerical calculations. Analytical methods are difficult to implement because of the structure complexity. Perhaps a unique example of the analytical approach to a circulator design is given in [9], where the authors used the temporal coupled-mode theory (TCMT) for the analysis of Y-circulator in PhC waveguide with an MO resonator. The threefold rotational symmetry

M

Manuscript received April 27, 2017; revised July 24, 2017 and October 4, 2017; accepted November 1, 2017. Date of publication December 13, 2017; date of current version March 5, 2018. This work was supported by the Brazilian agencies National Counsel of Technological and Scientific Development (CNPq) and the Coordination for the Improvement of Higher Education Personnel (CAPES). (Corresponding author: Gianni Portela.) The authors are with the Faculty of Electrical Engineering, Federal University of Pará, Pará, Belém 66075-110, Brazil (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2777981

Fig. 1. Three-port circulators with different types of magnetic symmetry and their elements of symmetry. Rectangles are waveguides, dotted circles denote MO resonators, H0 is dc magnetic field, and T σ is antiplane of symmetry. Circulators with threefold rotational symmetry are (a) Y-circulator with symmetry C3v (C3 ) [11] and (b) circulator with symmetry C3 [9]. Circulators described by magnetic group Cs (C1 ) are (c) T-circulator [2] and (d) W-circulator [1].

of the structure simplified greatly the analysis. The TCMT description of this circulator is based on the usual geometrical symmetry elements of rotation by 2π/3 and −2π/3 [see Fig. 1(a) and (b)]. In magnetic symmetry, they are called unitary elements. Some of the recently suggested three-port circulators are based on low-symmetry 2-D PhC junctions. These circulators can be of W- or T-format [1], [2] [see Fig. 1(c) and (d)] and also octopus- or fork-types [3]. All of them do not possess threefold rotational symmetry. However, they are characterized by another symmetry, namely, by an element known as the antiplane of symmetry [10]. In magnetic symmetry, it is called antiunitary element and presents a combination of the geometrical plane of symmetry and the time reversal operator. Notice that some electromagnetic devices can have symmetry described by both unitary and antiunitary elements.

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These PhC circulators with low symmetry have been analyzed by using matrix methods and by numerical computer programs. In comparison with analytical methods, matrix methods do not give much insight into the physics of the device as well as the numerical methods. Besides, the numerical methods have another drawback, namely, for the analysis of a given structure in a wide range of parameters and frequencies, these methods require much time and memory, i.e., they are costly. This is aggravated in optimization processes. The TCMT is one of the popular approximate analytical methods in the theory of resonance systems with weak decay rates [12]. It is based on some general physical principles, such as conservation of energy, time-reversal symmetry, and geometrical symmetry. In this paper, we shall remove the restriction of the time reversal symmetry. In order to simplify our discussion, we have chosen a W-circulator based on 2-D PhC as an example [Fig. 1(d)]. This circulator has a rather complex structure consisting of three waveguides and two resonator modes and, therefore, the theory presented in Sections II and III can be easily adapted to the structures with another number of ports and modes. Besides, this analysis allows one to see some resemblances and differences of the TCMT for the components described by only unitary elements of symmetry [9] with our case, where the circulator is defined only by an antiunitary element of magnetic symmetry, and to compare the results. We start with the related scattering matrix eigenvalue problem from the point of view of space–time reversal symmetry. The properties of the eigenvalues and eigenvectors of physical structures, which are described by magnetic groups, are closely related to the special properties of the time reversal operator. Then, we compare our analytical results with those obtained by a numerical method and discuss the application of a general TCMT method to some other devices and to the devices described by other magnetic symmetries. II. E IGENSOLUTIONS FOR I DEALLY M ATCHED L OSSLESS C IRCULATOR W ITH T σ S YMMETRY The three-port circulator under consideration [Fig. 1(d)] is magnetized by a dc magnetic field H0 oriented along the z-axis. The circulator is described by the magnetic group Cs (C1 ) (in Schoenflies notations [10]) with the elements e (unit element) and the combined element T σ , where T is the time reversal operator and σ defines the plane of symmetry. Notice that we use in this paper the so-called restricted time reversal operator [10], which preserves the passive or active nature of media. The 3-D representation of the operator σ , which interchanges ports 2 and 3, is ⎛ ⎞ 1 0 0 Rσ = ⎝ 0 0 1 ⎠. (1) 0 1 0 To calculate the structure of the scattering matrix S, one can use the commutation relation for the antiunitary element T σ , namely, Rσ S = S T Rσ [10], where T denotes transposition. As a result, the antiplane T σ defines the following structure

of this matrix:

⎛

S11 S = ⎝ S13 S12

S12 S22 S32

⎞ S13 S23 ⎠ S22

(2)

where S33 = S22 , S21 = S13 , and S31 = S12 . Using the unitary constraint SS † = S † S = I , where I is 3 × 3 unit matrix and † denotes Hermitian adjoint, the scattering matrix S, the eigenvalues s0 , s+ , and s− , and the corresponding normalized eigenvectors V0 , V+ , and V− for the ideally matched lossless three-port circulator with antiplane of symmetry can be easily calculated. Given the eigenvalue problem SV = sV and the solution of the characteristic equation det(S − s I ) = 0, one can define the aforementioned quantities for the direction of circulation (1 → 3 → 2 → 1) as follows: ⎞ ⎛ 0 1 0 S = ⎝ 0 0 eiϕ ⎠ (3) 1 0 0 s = ei(ϕ+2π)/3 s− = ei(ϕ−2π)/3 (4) s0 = eiϕ/3 ⎞ ⎛ + ⎞ ⎛ 1 1 1 1 V+ = √ ⎝ ei(ϕ+2π)/3 ⎠ V0 = √ ⎝ eiϕ/3 ⎠ 3 e−iϕ/3 3 e−i(ϕ+2π)/3 ⎞ ⎛ 1 1 ⎝ i(ϕ−2π)/3 ⎠ e V− = √ (5) 3 e−i(ϕ−2π)/3 where ϕ = ϕ23 − ϕ12 , ϕ23 , and ϕ12 are phases of the elements S23 and S12 , respectively. The eigensolutions (5) and (5) with the subindexes + and − correspond to the counter-rotating excitations of the junction. In formulas (3)–(5), the phase angle ϕ is defined not only by the geometrical angle ψ (see Fig. 1) but it depends also on the physical parameters of the three ports, in particular, on the parameters of the MO resonator. With ϕ = 0, solutions (3)–(5) are transformed into corresponding equations for Y-circulator with threefold rotational symmetry [11]. In the case of threefold symmetry C3v (C3 ) [Fig. 1(a)] or C3 [Fig. 1(b)], the element C3 strictly defines the ψ = 120° geometrical angle between the two ports and, as a consequence, ϕ = 0. In this case, the phase shifts between all the three ports are equal. However, in general symmetry description of three-port junctions with the magnetic symmetry T σ , the geometrical angle ψ between ports 2 and 3 is not defined [and it does not enter in the equations of TCMT as well (see in the following]. It may be, for example, 180° as in T-circulator [2] or 240° as in W-circulator [1] [see Fig. 1(d)]. It may also be 0° and 360° as in the fork- and octopus-type, respectively [3]. Therefore, the phase angle ϕ in the abovementioned equations, considering only symmetry arguments, is also not defined. Notice that the condition ϕ = 0 in (3) can be, in principle, fulfilled in a circulator without threefold rotational symmetry by choosing physical and geometrical parameters of the junction. However, this condition, which is not related to a geometrical symmetry, is rather difficult to achieve and, besides, characteristics of the component in this case are much sensitive to frequency changes. The above-mentioned discussion is closely related with the theory of irreducible representations (IRREPs). For the

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common plane of symmetry σ , there exist two IRREPs, namely, IRREP A and IRREP B, corresponding to even and odd solutions (see Table I of the Appendix). However, for the antiplane of symmetry (see Table II of the Appendix), there are corepresentations for the T σ element, namely, eiθ and −eiθ (which are equivalent) where the angle θ is not defined by symmetry constraints. Thus, in this case, one cannot use a general method of separation of the possible solutions in the even and odd functions. The arbitrariness of the angle ϕ in the above description is related with that of θ in the corepresentations of T σ . After diagonalization of the matrix S by using the eigenvectors (5), one comes to the following relations: S11 = S22 = S33

1 = (s0 + s+ + s− ) 3

1 −iϕ/3 e (s0 + s+ e−i2π/3 + s− ei2π/3 ) 3 1 = eiϕ/3 (s0 + s+ ei2π/3 + s− e−i2π/3 ) 3

S12 = S31 = S13 = S21

1 i2ϕ/3 e (s0 + s+ e−i2π/3 + s− ei2π/3 ) 3 1 (6) S32 = e−i2ϕ/3 (s0 + s+ ei2π/3 + s− e−i2π/3 ). 3 With ϕ = 0, these relations are reduced to the known ones for the Y-circulator [11]. The equality of the reflection coefficients in (6), S11 = S22 = S33 [in (1) one has only S33 = S22 ], follows from our approach based on the eigensolutions calculated for the case of ideal matching (S11 = S22 = S33 = 0). Notice that Y-circulators with threefold rotational symmetry can also possess the antiplane of symmetry T σ [see Fig. 1(a), where in fact the circulator has the three antiplanes T σi , i = 1, 2, 3]. However, in this case, it is sufficient to use only the unitary element C3 because from the point of view of scattering matrix, T σi does not give any additional information. S23 =

III. T EMPORAL C OUPLED -M ODE T HEORY A. General Description A general approach in TCMT for reciprocal structures is discussed in [13].In the following, we shall develop TCMT for the devices with T σ symmetry, describing the necessary modifications due to specific symmetry of the component and its nonreciprocity. The discussed circulators can usually work in two different regimes. First, the traditional regime based on a sum of two counter-rotating modes in the MO resonator possessing different frequencies. In the second regime, the circulation is provided by only one of the counter-rotating modes. Here, we shall consider the first regime with two resonant modes. For definiteness, we apply to a W-circulator in PhC with triangle lattice [1]. In Dirac’s bracket notations, the TCMT equations for our system with two counter-rotating orthogonal modes and three ports can be written as follows [13]: da = (i  − )a + K T |sin  dt |sout  = C|sin  + Da

(7) (8)

where a is the vector of cavity resonances, |sin  is the incoming wave, and |sout  is outgoing wave. They are, respectively, ⎛ ⎛ ⎞ ⎞   sin1 sout1 a+ a= |sin  = ⎝ sin2 ⎠ |sout  = ⎝ sout2 ⎠. (9) a− sin3 sout3 In (7),  and  are 2 × 2 Hermitian matrices, with  defining resonant frequencies and  describing the decay rates due to coupling of the resonances with the waveguides. These matrices can be written as     ω+ 0 γ+ 0 = = . (10) 0 ω− 0 γ− The 2 × 3 matrices K and D describe, respectively, coupling of three incoming waves with two resonant modes in (7), and two resonant modes with three outgoing waves in (8). On the other hand, the 3 × 3 matrix C in (8) is related to the direct coupling that takes place between the incoming and outgoing waves [13]. The entries of matrix C will be defined in Section III-B. Now, we apply to the scattering matrix S of the circulator in the steady state, which is defined by the relation |sout  = S|sin . For the incident wave with frequency ω, it follows from (7) and (8): S = [C + D F −1 K T ]

(11)

where F = (i ωI − i  + ) is a 2 × 2 matrix and I is a 2 × 2 unit matrix. In the following discussion of the circulator properties, we suppose, in the same way as in [9], that the nonreciprocity of the device is defined only by the effect of the frequency splitting of the counter-rotating modes of the resonator and the coupling coefficients are reciprocal. B. Matrices C, D, and K There is no pure time reversal operator T in the magnetic group of symmetry of our system (as well as in any other type of circulators). Therefore, application of the time reversal changes the scenario in the TCMT. As a consequence, the usual TCMT approach for reciprocal structures [13] where the time reversal operator is used should be modified. In the time-reversed regime, the external dc magnetic field H0 changes its sign, the sense of rotations in  is changed and, as a result, the direction of circulation (1 → 3 → 2 → 1) is switched to the reverse one, i.e., to (1 → 2 → 3 → 1). The matrices K and D are defined by using energy conservation principle and by time reversal changes as follows: D † D = 2

(12a)

K∗ = D C D ∗ = −D ∗

(12b) (12c)

where ∗ denotes complex conjugation and the specific TCMT scattering matrix C in (8) follows from (12c): ⎛ ⎞ −1 0 0 C = ⎝ 0 −1 0 ⎠. (13) 0 0 −1

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S12 = S31   e−i2π/3 2 ei2π/3 = e−iϕ/3 + 3 1 + i (ω − ω+ )/γ+ 1 + i (ω − ω− )/γ− (16) S13 = S21   ei2π/3 2 iϕ/3 e−i2π/3 = e + 3 1 + i (ω − ω+ )/γ+ 1 + i (ω − ω− )/γ− (17)   −i2π/3 i2π/3 e 2 e S23 = ei2ϕ/3 + 3 1 + i (ω − ω+ )/γ+ 1 + i (ω − ω− )/γ− (18)   i2π/3 −i2π/3 e 2 e S32 = e−i2ϕ/3 . + 3 1 + i (ω − ω+ )/γ+ 1 + i (ω − ω− )/γ− (19) With ϕ = 0, the above-mentioned formulas are reduced to those for the Y-circulator with threefold rotational symmetry [9]. D. T σ Symmetry

Fig. 2. Excitation of W-circulator by (a) eigenvector V+ with reflection coefficient s+ and (b) eigenvector V− with reflection coefficient s− .

Relation (12a) is proved in [13], and equality (12b) is written in [9]. However, (12c) differs from the corresponding relation presented in [13] for reciprocal components because its proof depends on the relationship between K and D matrices. The operators D and K describe the processes, which can be considered as going on in opposite directions in time. As a consequence, for reciprocal devices, one should consider K = D, which in turn results in the relation C D ∗ = −D presented in [13]. On the other hand, (12c) for nonreciprocal components follows from the fact that, in this case, the relation K ∗ = D is valid, since the operator T fulfills complex conjugation. Matrix D is defined by the eigenvectors V+ and V− (5) and also by . Using the schemes of excitation of the junction by the eigenvectors V+ and V− with the known reflection coefficients s+ and s− corresponding to SV± = s± V± (see Fig. 2) and the equation for the TCMT scattering matrix (11), one can show that at the frequency ω =  − i  ⎞ √ √  ⎛ γ+ γ− 2⎝ √ √ γ+ e i(ϕ+2π)/3 γ e i(ϕ−2π)/3 ⎠. (14) D= √ − −i(ϕ−2π)/3 3 √ −i(ϕ+2π)/3 γ+ e γ− e C. Scattering Matrix Elements From (11), one can obtain the S matrix elements S11 = S22 = S33 2 = −1 + 3



 1 1 + 1 + i (ω − ω+ )/γ+ 1 + i (ω − ω− )/γ− (15)

Now we analyze the effect of the combined operator T σ on the quantities of our problem. The following quantities are invariant with respect to T σ : H0 , S, V0 , V+ , and V− . The matrix (operator) D (14) must also be invariant with respect to the operator T σ , that is T σ D = D where the operator T fulfills complex conjugation. Application of the operator σ leads to the junction with ports 2 and 3 (i.e., line 2 and line 3 of the matrix D) interchanged. As a result, applying T σ to (14), one comes to T σ D = Rσ D ∗ = D

(20)

where Rσ is given by (1). The same is true for the operator K , i.e., T σ K = K . Besides, T σ  =  because the operator σ changes the sense of rotation as well as T does and, as a result, T σ preserves the sense of the eigemodes ω+ and ω− rotation. The effect of the operator T σ on the incoming and outgoing ∗ , T σ |s  = |R s ∗ . waves is as follows: T σ |sout  = |Rσ sin in σ out Using this analysis and also (7) and (8), one can show that if there is a solution |sout  for the excitation |sin , there exists ∗  with |R s ∗  as the excitation. another valid solution |Rσ sin σ out IV. I LLUSTRATION AND D ISCUSSION For illustration of the theory application, we have chosen the W-circulator shown in Fig. 3 [1]. It consists of a 2-D PhC formed by a triangle lattice of holes, with radius r = 0.3a, filled with air and etched in an MO material. The lattice constant a of the 2-D PhC can be adjusted according to the desired operating frequency band. For operation around the free-space wavelength λ = 1.55 μm, the lattice constant of the PhC is a = 480 nm. Bismuth–Iron–Garnet [14], [15] is the employed MO material and it is characterized by the magnetic permeability μ = μ0 and the electric permittivity tensor ⎛ ⎞ r −ig 0 r 0⎠ (21) [ ] = 0 ⎝ ig 0 0 r

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Fig. 3.

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PhC-based W-circulator [1].

Fig. 5. (Color online) Magnetic field component Hz of waves for excitation of different ports in W-circulator at the central frequency ωa/2π c = 0.3057 [1]. H0 is the external dc magnetic field and 1, 2, and 3 are the input/output ports.

Fig. 4. Frequency response of W-circulator for excitation of port 1 calculated by COMSOL and TCMT equations.

with r = 6.25 and g = 0.3. It is worth noticing that the off-diagonal parameter g of tensor [ ] is proportional to the applied dc magnetic field H0 . As can be seen from Fig. 3, three waveguides are produced in the PhC structure, by means of the removal of some holes in a straight line. Besides, one resonant cavity is formed by adjustment of the positions and radii of some holes located in the center part of the device. More details regarding the device geometry are provided in [1]. Here, we are mainly concerned with the development of the TCMT-based description, but not with the device design. In order to validate the presented TCMT description of the device, we have performed some computational simulations using the software product COMSOL Multiphysics and compared the results. The theoretical results, obtained with formulas (15)–(19), and the simulation results are shown in Fig. 4. We are presenting only the results for the case in which excitation is applied at port 1, since similar results can be obtained for the excitation of ports 2 and 3. One can see in Fig. 4 good agreement between the computational simulations and the theoretical results provided by the TCMT equations. The parameters of the TCMT description ω+ , ω− , γ+ , and γ− have been obtained from computational simulations of the isolated cavity, i.e., with no waveguides connected to it. More specifically, we have determined their values from the calculation of the frequency

splitting of the counterrotating modes, excited in the isolated cavity, as a function of parameter g. Their values are: ω+ = 1.1988 × 1015 rads−1 , ω− = 1.2004 × 1015 rads−1 , and γ+ = γ− = 1.388 × 1012 rads−1 . Similar to Y-circulator [9], the extremum of the parameter |S31 |2 , i.e., the maximum of the transmission power from port 1 to port 3 occurs at a frequency ω0 lying between the resonant frequencies ω− and ω√ + and satisfying the condition √ ω0 = ω− −γ− / 3 = ω+ +γ+ / 3. At the same frequency ω0 , one can observe also the maximum of the isolation of port 2 from port 1. The magnetic field component Hz of the waves for excitation of different ports in W-circulator at the central frequency ωa/2πc = 0.3057 (c is the speed of light in free-space) is shown in Fig. 5. The standing wave dipoles in Fig. 5(a) and (b) are oriented by 60° with respect to that in Fig. 5(c). The frequency band of this type of circulators at the level of isolation −15 dB is defined by |ω+ − ω− | g

f = 0.65 = 0.02 . f0 ω0 r

(22)

For small values of the Voigt parameter g/ r , the frequency splitting |ω+ − ω− | is proportional to g/ r [3]. Thus, the frequency band can be defined in terms of the Voigt parameter. In our case, the bandwidth of the circulator is 0.0831% for excitation applied in ports 1 and 3 and 0.0985% for excitation applied in port 2. V. P HASE -F REQUENCY C HARACTERISTICS OF W-C IRCULATOR In phase-sensitive applications, such as, for example, phased array antennas, it is important to know phase-frequency

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VI. C ONCLUSION We suggested a method of analysis of devices with MO resonators based on TCMT. The main peculiarity of our theory is the use of antiunitary elements of symmetry. The TCMT discussed in this paper can be applied to the structures with complex resonator systems (for example, with coupled resonators) and/or complex waveguides, where traditional analytical methods do not work. In these cases, one can consider separately the resonators and the waveguides theoretically or experimentally, because these separated problems are much easier to deal with and then combine the results. Notice that the operator T σ is occurred not only in different types of three-port circulators discussed here but also in some other nonreciprocal and control components with MO materials, such as switches, dividers, and multifunctional devices. In these cases, one can use the general theory developed in this paper. The described procedure can be used also for electromagnetic devices described by other antielements of symmetry, such as antirotation T Cn (n = 2, 4, 6, . . .) and anticenter of symmetry T i . A PPENDIX

Fig. 6. (a) Three-port W-circulator described by magnetic group Cs (C1 ). (b) Frequency dependence of phase angle ϕ = ϕ23 − ϕ12 .

responses of the components. In the case of Y-circulator, such analysis is fulfilled in [16]. Here, we discuss some peculiarities of these characteristics for W-circulator. First, one should note the principal difference in phasefrequency characteristics of W-circulator and Y-circulator. If the positions d of reference planes (2-2) and (3-3) in Fig. 6(a) are fixed symmetrically with respect to the antiplane T σ and the position c of reference plane (1-1) is chosen properly, one can provide ϕ = ϕ23 − ϕ12 = 0. However, the equality ϕ = 0 is fulfilled only at one fixed frequency. This is similar to the case which is called in the group-theoretical language “accidental degeneracy,” i.e., it is not defined by symmetry of the structure. In contrast to this, for the Y-circulator with the threefold rotational symmetry with symmetrical positions of the reference planes, the equality ϕ = 0 exists at any frequency. Besides, from Fig. 5(c), one can see that the phase shift ϕ23 for the transmission (3 → 2) defined on the circumference of the central hole is about π, while for the cases (1 → 3) [Fig. 5(a)] and (2 → 1) [Fig. 5(b)], it is approximately zero. Second, the TCMT described previously does not allow one to calculate the frequency dependence of ϕ. This should be done by another method. The characteristic ϕ(ω) for the discussed W-circulator calculated by COMSOL Multiphysics is presented in Fig. 6(b). One should note that, in the frequency range of the circulator ωa/2πc = [0.3056, 0.3058], defined at the isolation level −15 dB, the phase angle ϕ is a linear function of frequency, with the change of about 5° on this parameter.

For illustration and comparison purposes, we give a brief account of two simplest cases of groups in the following: one group describing a plane σ and the other one an antiplane T σ of symmetry. The former is frequently met in the reciprocal components analyzed by TCMT, and the latter is used in this paper. The group Cs contains the unit elements e and the plane σ . In fact, the full magnetic group of symmetry of an object without magnetic media contains also the time reversal operator T and the product of T with σ , i.e., T σ . Thus, it has four elements. But usually, the operator T is taken into account separately by imposing the requirements of symmetry of the scattering matrix and the material tensors of permittivity and permeability μ. The operator T σ is not considered at all, because it does not give any additional information. The IRREPs of the group Cs with two elements e and σ are given in Table I. These representations can be obtained from the multiplication scheme of the elements, i.e., σ 2 = e. In terms of IRREPs R (i.e., numbers), the eigenvalue equation for the element Rσ is Rσ2 = 1. Therefore, one has R A = 1 for representation A and R B = −1 for representation B (see the third column of Table I). These two IRREPs correspond, respectively, to the even and odd solutions with respect to the plane σ . On the other hand, the magnetic group Cs (C1 ), possessing the two elements e and T σ , can be considered as a subgroup of the group Cs with four elements described previously. The IRREPs (which are called corepresentations) of this group are written in Table II. The element T σ is antiunitary and antilinear one [17]. According to the basic properties of the group elements, the product of T σ with itself must be equal to the unit element, i.e., (T σ )(T σ ) = e. From Wigner’s theory of corepresentations [17] which defines, in particular, specific properties of the multiplication scheme for the antiunitary elements, one has RT σ RT∗ σ = 1,

DMITRIEV et al.: TCMT OF ELECTROMAGNETIC COMPONENTS

TABLE I IRREPs OF THE G ROUP Cs

TABLE II I RREDUCIBLE C OREPRESENTATIONS OF THE G ROUP Cs (C1 )

where the complex conjugation appears in the second term of the representation product (notice that the term corepresentation originates from the complex conjugation). Therefore, for the antiplane T σ , the two corepresentations c A and cB in Table II are Rc A = eiθ (i is the imaginary unit) and RcB = −eiθ , respectively, where θ is any real number. These two corepresentations are equivalent, because they are related by P −1 Rc A P ∗ = RcB , where P = i . This is very different from the case of the common symmetry plane σ . Here, in particular, it is impossible to describe a solution as a sum of two orthogonal members with even and odd symmetry with respect to the antiplane T σ . The theory of representations (reducible and irreducible) applied to electromagnetic structures is discussed, for example, in [18] and the theory of corepresentations in [10].

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[11] J. Helszajn, Nonreciprocal Microwave Junctions and Circulators. Hoboken, NJ, USA: Wiley, 1975. [12] H. Haus and W. P. Huang, “Coupled-mode theory,” Proc. IEEE, vol. 79, no. 10, pp. 1505–1518, Oct. 1991. [13] W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron., vol. 40, no. 10, pp. 1511–1518, Oct. 2004. ´ [14] W. Smigaj, J. Romero-Vivas, B. Gralak, L. Magdenko, B. Dagens, and M. Vanwolleghem, “Magneto-optical circulator designed for operation in a uniform external magnetic field,” Opt. Lett., vol. 35, no. 4, pp. 568–570, 2010. [15] Z. Wang and S. Fan, “Suppressing the effect of disorders using time-reversal symmetry breaking in magneto-optical photonic crystals: An illustration with a four-port circulator,” Photon. Nanostruct.-Fundam. Appl., vol. 4, no. 3, pp. 132–140, 2006. [16] J. Helszajn, W. D’Orazio, and M. Caplin, “Insertion phase and phase slope parameter of microwave junction circulators,” Proc. IEE—Microw., Antennas Propag., vol. 151, no. 1, pp. 54–60, 2004. [17] E. P. Wigner, Group Theory and Its Applications to the Quantum Theory of Atomic Spectra. Orlando, FL, USA: Academic, 1959. [18] X. Zheng et al., “On the use of group theory in understanding the optical response of a nanoantenna,” IEEE Trans. Antennas Propag., vol. 63, no. 4, pp. 1589–1602, Apr. 2015.

Victor Dmitriev received the M.Eng. and Ph.D. degrees in electrical engineering from the Bauman Moscow State University, Moscow, Russia, in 1971 and 1977, respectively. He has authored or co-authored over 200 conference papers, 110 journal papers, 2 books, and 8 book chapters. His current research interests include group theoretical methods in electromagnetic theory, the propagation of electromagnetic waves in complex media, metamaterials, antennas and nanoantennas, photonic crystals, nanoelectronics, and nanophotonics.

R EFERENCES [1] V. Dmitriev, M. N. Kawakatsu, and F. J. M. de Souza, “Compact three-port optical two-dimensional photonic crystal-based circulator of W -format,” Opt. Lett., vol. 37, no. 15, pp. 3192–3194, 2012. [2] Q. Wang, Z. Ouyang, K. Tao, M. Lin, and S. Ruan, “T-shaped optical circulator based on coupled magneto-optical rods and a side-coupled cavity in a square-lattice photonic crystal,” Phys. Lett. A, vol. 376, no. 4, pp. 646–649, 2012. [3] V. Dmitriev, G. Portela, and L. Martins, “Three-port circulators with low symmetry based on photonic crystals and magneto-optical resonators,” Photon. Netw. Commun., vol. 31, no. 1, pp. 56–64, 2016. [4] Z. Wu, M. Levy, V. Fratello, and A. M. Merzlikin, “Gyrotropic photonic crystal waveguide switches,” Appl. Phys. Lett., vol. 96, no. 5, p. 051125, 2010. [5] A. Esmaieli and R. Ghayour, “Magneto-optical photonic crystal 1 × 3 switchable power divider,” Photon. Nanostruct.-Fundam. Appl., vol. 10, no. 1, pp. 131–139, 2012. [6] V. Dmitriev and G. Portela, “Multifunctional two-dimensional photonic crystal optical component based on magneto-optical resonator: Nonreciprocal two-way divider-switch, nonreciprocal 120 deg bendingswitch, and three-way divider,” Opt. Eng., vol. 53, no. 11, p. 115102, 2014. [7] J. Helszajn, The Stripline Circulators: Theory and Practice, vol. 206. Hoboken, NJ, USA: Wiley, 2008. [8] S. Fan and Z. Wang, “An ultra-compact circulator using two-dimensional magneto-optical photonic crystals,” J. Magn. Soc. Jpn., vol. 30, no. 6_2, pp. 641–645, 2006. [9] Z. Wang and S. Fan, “Magneto-optical defects in two-dimensional photonic crystals,” Appl. Phys. B, Lasers Opt., vol. 81, nos. 2–3, pp. 369–375, 2005. [10] A. Barybin and V. Dmitriev, Modern Electrodynamics and CoupledMode Theory: Application to Guided-Wave Optics. Princeton, NJ, USA: Rinton Press, 2002.

Gianni Portela received the B.S. degree in mathematics from Pará State University, Pará, Brazil, in 2006, and the B.S., M.Eng., and Ph.D. degrees in electrical engineering from the Federal University of Pará, Belém, Brazil, in 2007, 2008, and 2015, respectively. He is currently a Post-Doctoral Fellow with the Federal University of Pará. His current research interests include the development of photonic crystals-based devices, such as switches, dividers, circulators, isolators, and multifunctional components.

Leno Martins received the B.S. degree in computer engineering and M.Eng. degree in electrical engineering from the Federal University of Pará, Pará, Belém, Brazil, in 2013 and 2016, respectively, where he is currently pursuing the Ph.D. degree. His current research interests include the development of photonic crystal-based circulators.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 3, MARCH 2018

A Short–Open Calibration Method for Accurate De-Embedding of 3-D Nonplanar Microstrip Line Structures in Finite-Element Method Yin Li , Student Member, IEEE, and Lei Zhu, Fellow, IEEE

Abstract— This paper presents the numerical short–open calibration (SOC) method in the full-wave finite-element method (FEM) algorithm for accurate de-embedding of equivalent circuit parameters of a variety of 3-D nonplanar periodic guided-wave and discontinuity structures. It holds its attractive features in high efficiency and good accuracy as demonstrated in the method of moments algorithm, and further allows itself to be capable of characterization of 3-D nonplanar structures. By setting the reference plane along each feeding line as the perfect electric and magnetic conductors, i.e., PEC and PMC, the entire error box involved in each feeding line and impressed source can be characterized by using these two calibration standards, namely, short- and open-ended circuits. Formulation of ABCD matrix of this error box is described under the introduction of an impressed voltage at each port. Then, ABCD matrix of the core circuit structure can be effectively de-embedded or extracted. Finally, a few numerical examples are given to demonstrate the efficiency and accuracy of our proposed full-wave 3-D FEM-SOC approach in numerical de-embedding of 3-D nonplaner structures. Index Terms— Finite-element method (FEM), numerical calibration, port discontinuity, short–open calibration (SOC), 3-D nonplanar structure.

I. I NTRODUCTION

T

HE development of integrated and multifunctional microwave circuits has been arousing the rapidly growing demand for high-accuracy and high-efficiency modeling of 3-D planar or nonplanar circuit structure, called as device-under-test (DUT) in microwave measurement. To do it, the impressed source or excitation models are indispensable to be introduced at all the ports of these DUT in numerical algorithms. Usually, these sources at ports can be divided into two distinctive types: wave port and impressed port. Ideally, the so-called port discontinuity does not exist if the field distribution at a port can be exactly expressed in

Manuscript received June 22, 2017; revised September 21, 2017 and October 25, 2017; accepted November 12, 2017. Date of publication December 11, 2017; date of current version March 5, 2018. This work was supported in part by the University of Macau under Grant MYRG201700007-FST, Grant MYRG2015-00010-FST, and Grant CPG2017-00028-FST, in part by the Macao Science and Technology Development Fund under Grant FDCT/051/2014/A1, and in part by the National Nature Science Foundation of China under Grant 61571468. (Corresponding author: Yin Li.) The authors are with the Department of Electrical and Computer Engineering, Faculty of Science and Technology, University of Macau, Macao, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2778111

mathematical modeling. However, it is theoretically impossible to derive the analytical or exactly mathematical expression of the introduced field distribution, especially when the incident wave or impressed source is introduced at a port of an irregularly shaped feeding line or waveguide. In this case, the field distribution in the transverse plane needs to be numerically derived by solving an eigenvalue problem [1], [2]. Meanwhile, the field distribution at port can be determined by solving a set of expansion modal functions. However, it is very challenging to accurately fit the port field using a sufficient number of these modal functions, resulting in unwanted field mismatching or discontinuity problem at the port. The impressed sources (voltage or current sources) have been widely employed to express the field distribution, and many previous works have verified that these impressed sources can approximately represent the field at port [3]–[6]. However, the field distribution in the transverse plane at the port is basically inconsistent with its actual one, thus causing the so-called port discontinuity at the port with impressed source. In microwave measurement, it has been well reported that the error at a port between the connecting cable and feeding line can be effectively eliminated by different calibration methods, such as short–open–load–thru, thru–reflect–line (TRL), and thru–reflect–match [7]. As such, a few numerical calibration techniques have been proposed to de-embed the port discontinuity in numerical algorithms, which is undesired and nonphysical. The so-called “double-delay de-embedding” was developed to de-embed the sole port discontinuity aroused by a delta-gap voltage source [8]–[10]. In this context, the port discontinuity is equivalently expressed as frequencydependent shunt capacitance and/or inductance. This port model needs to simply predefine the lumped-element model of the complex, unwanted, and nonphysical port discontinuity. Consequently, it is not effectively workable for all the cases, especially at high frequency, where the dielectric substrate is not electrically thin. Then, the short–open calibration (SOC) technique was first introduced in 1997, and this method can remove all the unwanted parasitic effect brought by the approximation of the impressed source [11]. It has been implemented with the full-wave method of moment (MoM) algorithm to extract the equivalent model parameters of planar circuit/discontinuity elements involved in a complicated layout [12]–[14].

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LI AND ZHU: SOC METHOD FOR ACCURATE DE-EMBEDDING OF 3-D NONPLANAR MICROSTRIP LINE STRUCTURES

Meanwhile, it has been explored for parametric de-embedding of various nonuniform transmission lines with periodic configuration [15], [16]. Later on, an extended SOC method was utilized for modeling of multimode coplanar waveguide (CPW) [17], and a vector SOC method was developed for modeling of multiport circuits [18]. Due to its unique property in numerical de-embedding procedure, it has gained wide application in de-embedding or parametric extraction of various planar microwave circuits/discontinuities [19], [20], transmission line metamaterials on CPW [21], spoof surface plasmons [22], and so on. However, they were only focused on modeling of 2-D or 2.5-D planar structures and they are invalid for 3-D nonplanar structures. Thus, more calibration methods are needed to numerically handle more complicated 3-D nonplanar microwave circuits. Recently, the short–open–load (SOL) calibration method was proposed to extract the propagation constant and characteristic or wave impedance of the substrate-integrated waveguide (SIW) [23], [24] by virtue of the short, open, and load standards in the 3-D commercial full-wave simulator. However, the port impedance must be known or calculated in advance. In this context, a multimode thru–thru technique was proposed in [25] to extract the characteristic impedance of SIW operating at different modes. But, this method unfortunately preinquires the information of the wave port along its respective feed line [25]. The numerical TRL is another calibration method used to accurately extract the circuit discontinuity and structure, but it needs a prior knowledge of the characteristic impedance at the port or reference plane [25]–[27]. A similar numerical calibration method via a thru–line was proposed to derive the effective propagation constant of SIW with periodic vias in two sides by solving the eigenvalue problems [28], [29], but it cannot be applied to extract its effective characteristic impedance. As such, these problematic issues have restricted their applications in all of these numerical calibration techniques. The SOC method can proceed our de-embedding procedure without knowing such as port impedance, and can be applied to different feed networks with various sources models such as impressed voltage source, impressed current source, and wave port. This method can accurately extract all the effective per-unit-length transmission parameters of a periodic guidedwave structure, including the complex propagation constant and complex characteristic impedance. Moreover, it can be implemented in the 3-D volume integral equation [30] method for modeling of 3-D microelectromechanical systems. It is implemented by using the time-domain adaptive integral method for analysis of packaged geometries [31]. In this paper, the SOC method will be implemented in the 3-D finite-element method (FEM) algorithm for accurate modeling and de-embedding of 3-D nonplanar structures and circuits with complicated geometry. After the FEM algorithm is briefly described to model a generalized circuit structure with external ports, the two calibration standards, namely, short and open circuits, are formulated and characterized in the same FEM algorithm. As the port discontinuities are all known and further calibrated out, the core DUT portion can be effectively de-embedded. In final, a few examples are

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given to evidently validate the effectiveness of the proposed FEM-SOC approach in numerical de-embedding of 3-D nonplanar structures. II. FEM-SOC F ORMULATION A. Formulation of Edge-Based FEM and Port Model Based on the well-known Maxwell’s equations, the inhomogeneous vector wave equation can be obtained as [1]   (1) ∇ × μr−1 ∇ × E(r) − k0 r E(r) = − j k0η0 Jinc where Jinc is the electric current density, k0 the free-space wavenumber, η0 the free-space wave impedance, and r and μr the relative permeability and relative permittivity, respectively. According to the variation principle, a function F can be defined by an interproduct of Ea on the left-hand side of (1)    Ea · μr−1 (∇ × E) − k02 r E · d V F(Ea , E) = V  Ea · Jinc d V. (2) + j k0η0 V

In general, the FEM can be formulated and derived by using Galerkin’s method or the Ritz method [1], and it should be noted that the former one is chosen in our formulation to ensure the accuracy in numerical simulation. By choosing the expansion functions as the basis and weighting functions, the unknown electric field intensity in the whole 3-D volume can be in general expanded as E=

N 

Nie E ie = {E e }T {N e }

(3)

i=1

where Nie is the edge-based vector basis functions in the form of tetrahedral elements [1], {E e } is the unknown coefficients of the basis functions, and N is the number of the unknowns. By substituting (3) into (2), we can deduce  1  e T {E } [A]{E e } − k02 {E e }T [B]{E e } + j k0[b]{E e } . 2 N

F=

e=1

(4) Upon carrying out the summation and using the global notation, (4) can be written as a matrix system, such that   (5) [A] − k02 [B] {E} = j ωμ0 [b] where the matrix A, B, and b can be mathematically expressed as    ∇ × Nie (r) · μr−1 ∇ × Nej (r) d V (6) Ai j = 

Ve



Ve

Bi j = bi j =

Ve

  k02 r Nie (r) · ∇ × Nej (r) d V

(7)

Nie (r) · Jincd V .

(8)

Ideally, the current source model is composed of an infinite thin current inserted between two separated conductors, i.e., strip conductor and ground plane, at the feed position. This impressed current density Jimp can be mathematically

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Fig. 1. Schematic of the impressed current source model in the FEM algorithm.

expressed as a delta function in the vertical axis at the feeding point, as shown in Fig. 1, that is, Jimp(r) = yˆ I0 δ(x − x 0 , z)

(9)

where yˆ stands for to the direction of the impressed current source along the y-axis, and I0 represents its magnitude. This simplified current source in (9) can be applied to excite the dominant quasi-TEM mode and other higher order modes in the microstrip feeding line, especially at high frequency. As such, this port model inevitably brings out the unexpected error at the port and it should be calibrated out of the resultant matrix. B. Implementation of SOC Method in FEM Algorithm When the network parameters are numerically de-embedded by FEM method, the field matching between the external port and the interior feeding-line region should be guaranteed at first. It means that the incident wave, traveling longitudinally from infinite location, causes no scattered waves from the port to the outsides. To realize this condition, the perfect matched layer (PML) and absorbing boundary condition (ABC) need to be used to absorb the unexpected reflected wave. However, both of them cannot effectively absorb the evanescent waves at port over a wide frequency range. In this paper, such an error at each port, in other words, called as port discontinuity, is fully admitted by the reason of nonideal source. This effect has been entirely modeled in the same FEM algorithm and calibrated out of the resultant network matrix by virtue of two ideal calibration standards, i.e., short and open circuits. Fig. 2(a) shows the generalized two-port 3-D microwave structure with arbitrary shape and configuration. As indicated in Fig. 2(a), the entire structure can be decomposed into three distinctive parts: the two feeding lines with their impressed sources at two ports and the core 3-D nonplanar circuit block or DUT between the reference planes #1 and #2 . The two feeding lines are in general expressed as two respective error boxes, namely, [X 1 ] and [X 2 ], as shown in Fig. 2(b), and they include all the discontinuities caused by the nonideal lumped port model and numerical discretization of feedingline portion. By calibrating them out, the desired network parameters for the two-port DUT in Fig. 2 can be effectively de-embedded. In general, the i th error box can be expressed by the ABCD matrix with four terms ai , bi , ci , and di . For the i th port,

Fig. 2. Geometry and equivalent circuit representation of a generalized 3-D nonplanar microwave structure. (a) Physical geometry. (b) Entire equivalent model with three distinctive blocks, where two error boxes are relevant to two feed lines with external ports.

Fig. 3. Physical port models and equivalent circuit networks of SOC standards in SOC procedure. (a) Short standard. (b) Open standard.

the relationship between current/voltage and their respective ones at the reference plane is given by     Vi ai bi Vi = . (10) Ii ci d i Ii Herein, the ideal open and short ends are introduced at the reference plane of the i th feeding line, and the portion between the port and its respective reference plane is in general expressed as the error box [X i ] (i = 1, 2). As introduced in [12], the SOC technique defines a pair of calibration standards, i.e., ideal short and open circuits, and these two standards are formulated by placing an electric wall (E. W.) and magnetic wall (M. W.) at the reference plane of each feeding line. Meanwhile, the core DUT portion is defined between the reference planes #1 and #2 , as depicted in Fig. 3. According the electromagnetic theory, these short and open circuit standards can be exactly realized in the same FEM algorithm. In the open standard, the M. W. is realized with perfect magnetic conductors PMC boundary condition and the current at the reference plane is exactly equal to zero. Then, the port voltage (Vio ), port current (Iio ), and current at reference (Iis ) at the two terminals of the open standard should

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satisfy the relationship as the following equation: ai Vio + bi Iio = Vio

(11)

ci Vio + di Iio = 0.

(12)

As for the short standard, the voltage at the reference plane is exactly equal to zero and it is realized with the perfect electric conductor (PEC) boundary condition in simulation. Accordingly, its port voltage (Vis ) and current (Iio ) leads to the following equation: ai Vis + bi Iis = 0.

(13)

Since the two-port error boxes are reciprocal networks, the four elements of their respective ABCD matrix are related as ai di − bi ci = 1.

(14)

In fact, (11)–(14) are independent of each other, so the ABCD matrix of each error box can be obtained by solving these four equations once the current and voltage are derived. To simplify our formulation, the two-port voltages of short and open circuits, and one extra voltage at the reference plane are all normalized by the corresponding port current, such that ⎧ ⎪ ⎨V io = Vio /Iio  (15) V io = Vio /Iio ⎪ ⎩ V is = Vis /Iis . In this way, the ABCD matrix of error box [X i ] can be expressed as ⎤ ⎡ V io V is V io V is − ⎥ ⎢ V io − V is ⎥ ⎢ V io − V is (16) [X i ] = ⎢ ⎥  V io ⎦ ⎣ 1 −  V io V io where all the elements of the error box [X i ] is only related to the three equivalent normalized voltages. Once the ABCD matrices of these two error boxes are numerically derived, the ABCD matrix of the core circuit [ ADUT ] can be obtained by calibrating out these two error boxes from the entire network [AEXT ], such that [ ADUT ] = [X 1 ]−1 [ AEXT][X 2 ]−1 .

(17)

In numerical FEM algorithm, the PMC boundary condition is introduced at the port position to establish the impressed current source model, as shown in Fig. 2(a). In general, the PEC and PMC boundary conditions have zero tangential electric and magnetic fields nˆ × E = 0 nˆ × (∇ × E) = 0.

(18) (19)

In this context, the network parameters of the error boxes [X 1 ] and [X 2 ] can be modeled by the numerical FEM by virtue of the two standard elements with known ABCD matrices. In parallel, the actual or effective propagation constant γ and characteristic/wave impedance Z 0 of a guided-wave structure with uniform or periodically varied configuration can be derived without any prior assumption if its ABCD matrix with

Fig. 4. Numerical de-embedding of a microstrip line with finite length of L, which is excited by two feeding lines with the impressed port sources.

finite length of L can be numerically de-embedded from the above-described SOC procedure, such that a+d (20) 2 b Z0 = (21) c where L stands for the length of the core guided-wave structure as chosen in numerical modeling. cosh(γ L) =

III. N UMERICAL R ESULTS AND VALIDATION In this section, the SOC technique integrated with FEM simulator, namely, FEM-SOC approach, will be applied to numerically de-embed the finite-length microstrip line and finite-length microstrip line electromagnetic bandgap (EBG) structure in terms of their effective characteristic impedance and propagation constant as well as the two-layered via-hole microstrip line discontinuity in terms of its two-port equivalent circuit model over a wide frequency range. A. Microstrip Line Fig. 4 illustrates the schematic for numerical modeling of a finite-length microstrip line on a 1.27-mm-thick dielectric substrate with a relative permittivity of 10.2. Herein, the strip width is set as W = 1.04 mm, the length of the core microstrip line portion is selected as L = 30 mm, the width of ground plane is set as d = 10 mm, and the length between the port and its respective reference plane is selected as L = 10 mm. Fig. 5 shows the comparison among the propagation constants γ /k0 of the microstrip line calculated by FEM with and without the SOC technique, as well as the closed-form equation [32]. The results from the FEM-SOC agree well with its respective analytical results [32]. The relative error between them is less than 5% over a wide frequency range of 1–4 GHz. However, compared with the results from the FEM without SOC technology, the relative error turns to be larger than 15%. It is mainly caused by the fact that the parasitic port error is not calibrated out as extensively discussed in [12] and [14]. Fig. 6 illustrates the characteristic impedance of the microstrip line simulated by the FEM with or without SOC method. Its real part from the FEM without SOC is decreased as the frequency increases. When SOC is integrated in the

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Fig. 5. De-embedded propagation constant of the microstrip line with finite length of L by using different methods.

Fig. 7. Layout of the microstrip line EBG structure and its equivalent circuit representation. (a) Physical model with the current source at the ports of two feeding lines. (b) Equivalent circuit model for de-embedding of complex propagation constant and characteristic impedance using FEM-SOC.

Fig. 8. Mesh information of unit cell of periodic structure. (a) Front view. (b) Bottom view.

Fig. 6. De-embedded characteristic impedance of the microstip line with finite length of L by using different methods.

FEM simulator, the relevant result can remain stable as frequency is changed from 1 to 4 GHz, and it matches well with the analytical result. As for the imaginary part, all the three sets of results are all nearly zero because the losses from the dielectric substrate, conductor, and radiation are neglible. B. Microstrip Line Electromagentic Bandgap Structure Fig. 7(a) depicts the geometry of the finite-length microstrip line with N cell periodic backside aperture, named as the microstrip EBG structure, where the periodicity is set as T . This periodic structure is fed by the two uniform feeding lines with impressed current sources at the port planes #1 and #2. Moreover, the entire network is classified as the two error boxes in two sides and the core DUT or EBG with respect to the reference planes #1 and 2 , as illustrated in Fig. 7(b). The periodic FBG is characterized as an effective transmission line with length of L = N × T , complex characteristic impedance

of Z 0 = Z r + j Z i , and complex propagation constant of γ = α + jβ. The two error boxes indicate the entire effect of each feed line and the port discontinuity, and they can be effectively evaluated and calibrated out by using the proposed FEM-SOC. As a result, the ABCD matrix of the two-port network, representing the core EBG portion in middle, can be accurately extracted. Using (20) and (21), its effective complex characteristic impedance and complex propagation constant can be eventually derived. As indicated in Figs. 8 and 9, two per-unit-length transmission parameters of this periodic EBG structure extracted by the FEM-SOC agree well with those from the MoM-SOC [33]. The mesh information of a periodic unit cell is shown in Fig. 8. Fig. 9 illustrates the extracted phase constant β and attenuation coefficient α normalized by the free-space wavenumber k0 . As the frequency increases from 2 to 8 GHz, β/k0 rises up. Afterward, it falls down until around 12 GHz, and then keeps an upward tendency. The attenuation constant α/k0 almost keeps as zero in the whole frequency range, except in the bandstop range, in which it goes up and then down. Meanwhile, the real part of characteristic impedance Z r keeps an exponential growth as the frequency increases, then suddenly drops to zero, and keeps stable in bandstop range in final. From the upper of the bandstop range, it increases again as shown in Fig. 10(a). Its imaginary part Z i

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Fig. 9. De-embedded complex propagation constant of the microstrip line by using the FEM-SOC and MoM-SOC methods.

Fig. 10. De-embedded complex characteristic impedance of the microstrip line by using the FEM-SOC and MoM-SOC methods. (a) Real part. (b) Image part.

decreases quickly from a large value to zero over the bandstop range, as shown in Fig. 10(b). Fig. 11 shows the comparison of the results calculated by the FEM-SOC, and commercial software ANSYS high-frequency

Fig. 11. Comparison of the results of the microstrip line calculated by using the FEM-SOC and commercial software HFSS with two different port models. (a) Complex propagation constant. (b) Real part of complex characteristic impedance. (c) Imaginary part of complex characteristic impedance.

structural simulator (HFSS) with two different port models, called as wave port and lumped port, respectively. As shown in Fig. 11(a), the complex propagation constant from FEM-SOC is found to agree well with those from the wave port model in the HFSS. And the port discontinuity existing

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Fig. 12. Geometry of via-hole discontinuity in a two-layered substrate and its circuit model. (a) 3-D view. (b) Top view. (c) Equivalent circuit model.

Fig. 13. Frequency-dependent inductance L and capacitance C extracted from our FEM-SOC method under different geometrical dimensions. (a) Curves of L and C under different d1 . (b) Curves of L and C under different d2 .

based on a simple current source model without adding any PML ABC at all the ports. in current source used in FEM at each port needs to be fully removed. The lumped port model has been widely used in HFSS, and it is similar to the current source model in our proposed FEM-SOC method. Without removing this port discontinuity in FEM, the simulation results show unstable trend and incorrect value for the propagation constant, as shown in Fig. 11(a). Similarly, the derived complex characteristic impedance from the lumped port model in HFSS becomes unstable and incorrect again, as illustrated in Fig. 11(b) and (c). Therefore, the port discontinuity in the lumped port model needs to be fully solved and removed in all the FEM algorithms and commercial software. From this example, it has been confirmed that the proposed FEM-SOC method is powerful to efficiently and accurately extract the per-unit-length transmission parameters of the periodic guided-wave structures as realized in the MoM-SOC. Compared with these FEM algorithms or softwares under the lumped port model, the proposed FEM-SOC can directly and accurately obtain the results of a 2-D/3-D microwave circuit

C. Via-Hole Discontinuity in Two-Layered Substrate The via-hole is often used in the microwave multilayered structure such as the through-silicon via (TSV). It allows full transmission from one layer to another layer at low frequency and serves as a series inductive element at high frequency. A via-hole discontinuity across the two-layered substrate is depicted in Fig. 12(a) and (b). In the FEM-SOC modeling, its two feeding lines are excited by the impressed current sources at the two-port positions backed by the M.W. or PMC. The equivalent circuit model of the via-hole defined at the two reference planes #1 and #2 can be expressed as a π-network [34], and it is shown in Fig. 12(c). Fig. 13 depicts a few sets of frequency-dependent curves of the equivalent shunt capacitances and series inductances as extracted by our proposed FEM-SOC method. The extracted L and C values with respect to different geometrical parameters are illustrated in Fig. 12. In the frequency range from

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IV. C ONCLUSION In this paper, the FEM-SOC method is presented for accurate modeling and de-embedding of 3-D nonplanar microwave circuits and structures. By implementing the SOC technique in the FEM algorithm, the error box of each feeding line with impressed source at the port is accurately modeled and effectively calibrated out by virtue ideal short and open standards. As such, it allows us to efficiently and accurately de-embed or extract the core 3-D nonplanar circuit or structure of our interest. In final, three numerical examples are given to demonstrate and validate that the proposed FEM-SOC method is a powerful approach in modeling of 3-D nonplanar structures with complicated geometry. R EFERENCES Fig. 14. Geometry of a two-port circuit constituted by two cascaded viaholes and its equivalent circuit model. (a) Top view of its physical layout. (b) Equivalent circuit model.

Fig. 15. Two-port S-parameters of the cascaded via-hole circuit in Fig. 12(a), derived from its equivalent circuit model in Fig. 12(b) and direct simulation via HFSS.

1 to 4 GHz, it is apparent that the extracted L is mainly dependent on the inner radius d1 of the hole, while receiving almost no influence from varied d2 in the whole frequency band. Similarly, the simulated capacitance C is primarily dependent on the d2 , and keeps a relatively stable increment with respect to the varied d2 , as shown in Fig. 13(b). The inductance L decreases and the capacitance C increases as the frequency increases. Next, the two cascaded via-hole circuit in Fig. 14 is characterized by using the equivalent circuit model in Fig. 12(b) and direct simulation via HFSS. Both two sets of results are shown in Fig. 15 for comparison, and they are found in good agreement with each other over the whole frequency range. As a consequence, equivalent circuit model parameters of this via-hole discontinuity in Fig. 15 have been evidently validated, thereby evidently providing the verification on the effectiveness of the proposed FEM-SOC method again.

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[19] L. Zhu and K. Wu, “Field-extracted lumped-element models of coplanar stripline circuits and discontinuities for accurate radiofrequency design and optimization,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 4, pp. 1207–1215, Apr. 2002. [20] S. Sun and L. Zhu, “Guided-wave characteristics of periodically nonuniform coupled microstrip lines-even and odd modes,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 4, pp. 1221–1227, Apr. 2005. [21] J. Gao and L. Zhu, “Characterization of infinite- and finite-extent coplanar waveguide metamaterials with varied left- and right-handed passbands,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 805–807, Nov. 2005. [22] X. Liu, L. Zhu, Q. Wu, and Y. Feng, “Highly-confined and low-loss spoof surface plasmon polaritons structure with periodic loading of trapezoidal grooves,” AIP Adv., vol. 5, no. 7, p. 077123, 2015. [23] Q.-S. Wu and L. Zhu, “Numerical de-embedding of effective wave impedances of substrate integrated waveguide with varied via-to-via spacings,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 1, pp. 1–3, Jan. 2016. [24] L. Zhu, Q.-S. Wu, and S.-W. Wong, “Numerical SOC/SOL calibration technique for de-embedding of periodic guided-wave structures,” in Proc. IEEE Int. Conf. Comp. Electromag. (ICCEM), Feb. 2016, pp. 325–327. [25] F. Fesharaki, T. Djerafi, M. Chaker, and K. Wu, “S-parameter deembedding algorithm and its application to substrate integrated waveguide lumped circuit model extraction,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 4, pp. 1179–1190, Apr. 2016. [26] X.-P. Chen and K. Wu, “Accurate and efficient design approach of substrate integrated waveguide filter using numerical TRL calibration technique,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 1231–1234. [27] L. Li, K. Wu, and L. Zhu, “Numerical TRL calibration technique for parameter extraction of planar integrated discontinuities in a deterministic MoM algorithm,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, pp. 485–487, Dec. 2002. [28] F. Xu, K. Wu, and W. Hong, “Domain decomposition FDTD algorithm combined with numerical TL calibration technique and its application in parameter extraction of substrate integrated circuits,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 1, pp. 329–338, Jan. 2006. [29] L. Han, K. Wu, X.-P. Chen, and F. He, “Accurate analysis of finite periodic substrate integrated waveguide structures and its applications,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 864–867. [30] M. Farina and T. Rozzi, “A 3-D integral equation-based approach to the analysis of real-life MMICs application to microelectromechanical systems,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 12, pp. 2235–2240, Dec. 2001. [31] A. E. Yilmaz, J. M. Jin, and E. Michielssen, “Analysis of low-frequency electromagnetic transients by an extended time-domain adaptive integral method,” IEEE Trans. Adv. Packag., vol. 30, no. 2, pp. 301–312, May 2007. [32] D. M. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ, USA: Wiley, 2005, ch. 3, pp. 144–145. [33] L. Zhu, “Guided-wave characteristics of periodic microstrip lines with inductive loading: Slow-wave and bandstop behaviors,” Microw. Opt. Technol. Lett., vol. 41, no. 2, pp. 77–79, Apr. 2004. [34] T. Wang, R. F. Harrington, and J. R. Mautz, “Quasi-static analysis of a microstrip via through a hole in a ground plane,” IEEE Trans. Microw. Theory Techn., vol. 36, no. 6, pp. 1008–1013, Jun. 1988.

Yin Li (S’15) received the B.S. degree in applied physics from the China University of Petroleum, Dongying, China, in 2009, and the M.Eng. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2012. He is currently pursuing the Ph.D. degree at the University of Macau, Macau, China. From 2013 to 2015, he was a Research Assistant with the University of Hong Kong, Hong Kong. His current research interests include computational electromagnetic and microwave circuits.

Lei Zhu (S’91–M’93–SM’00–F’12) received the B.Eng. and M.Eng. degrees in radio engineering from the Nanjing Institute of Technology (now Southeast University), Nanjing, China, in 1985 and 1988, respectively, and the Ph.D. degree in electronic engineering from the University of ElectroCommunications, Tokyo, Japan, in 1993. From 1993 to 1996, he was a Research Engineer with Matsushita-Kotobuki Electronics Industries Ltd., Tokyo. From 1996 to 2000, he was a Research Fellow with the École Polytechnique de Montréal, Montréal, QC, Canada. From 2000 to 2013, he was an Associate Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. In 2013, he joined the Faculty of Science and Technology, University of Macau, Macau, China, as a Full Professor, where he served as the Head of Department of Electrical and Computer Engineering from 2014 to 2017 and has been a Distinguished Professor since 2016. He has authored or co-authored more than 395 papers in international journals and conference proceedings. His papers have been cited more than 4860 times with the H-index of 38 (source: ISI Web of Science). His current research interests include microwave circuits, guidedwave periodic structures, planar antennas, and computational electromagnetic techniques. Dr. Zhu was a recipient of the 1997 Asia–Pacific Microwave Prize Award, the 1996 Silver Award of Excellent Invention from Matsushita-Kotobuki Electronics Industries Ltd., and the 1993 First-Order Achievement Award in Science and Technology from the National Education Committee, China. He was an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES (during 2010–2013) and the IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS (during 2006–2012). He served as a General Chair of the 2008 IEEE MTT-S International Microwave Workshop Series on the Art of Miniaturizing RF and Microwave Passive Components, Chengdu, China, and a Technical Program Committee Co-Chair of the 2009 Asia–Pacific Microwave Conference, Singapore. He served as a member of the IEEE MTT-S Fellow Evaluation Committee (during 2013–2015), and has been serving as a member of the IEEE AP-S Fellows Committee (during 2015–2017).

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Modal Analysis and Propagation Characteristics of Leaky Waves on a 2-D Periodic Leaky-Wave Antenna Sohini Sengupta , David R. Jackson, Fellow, IEEE, and Stuart A. Long, Life Fellow, IEEE

Abstract— A 2-D periodic leaky-wave antenna is studied to examine the mechanism of beam formation from leaky-wave modes and the properties of the leaky-wave modes. The structure consists of a grounded dielectric slab with a periodic arrangement of narrow rectangular patches of metal on the upper surface of the dielectric, excited by a slot in the ground plane. The dimensions of the unit cell and the size of the patches determine the phase constant and the attenuation (leakage) constant of the leaky modes supported by the structure. The narrow beam that is formed is found to be due to an aperture distribution mainly consisting of the radiating space harmonics (Floquet waves) of two different leaky modes, one determining the E-plane behavior and one determining the H-plane behavior. The beamwidth is larger in the H-plane than in the E-plane due to the larger leakage constant of the H-plane mode. It is found that the two leaky modes also have different field distributions, in addition to having very different leakage constants. Index Terms— 2-D periodic leaky-wave antenna, 2-D periodic structure, dispersion behavior, floquet harmonics, grating lobes, leaky plasmon waves.

I. I NTRODUCTION

T

HE initial work in the area of 2-D leaky-wave antennas (LWAs) that are based on a partially reflecting surface (PRS) was done by Von Trentini [1], followed by Alexopoulos and Jackson [2], [3], Jackson and Oliner [4] and Jackson et al. [5]. A 2-D LWA using a periodic PRS was further explored using different kinds of elements [6], and the radiation properties of structures involving metal patch PRS elements were given in [7]. The PRS LWA topic has been studied extensively in recent years, including diverse contributions such as advances in the analysis of the structures [8], broadening the pattern bandwidth [9]–[14], reducing the height of the structure [15]–[17], array thinning [18], incorporation of metamaterials [19]–[21], designs for the mmwave region [22]–[26], and applications in the near-infrared regime [27]. A summary of some of the basic properties of 2-D

Manuscript received June 27, 2017; revised September 21, 2017 and November 1, 2017; accepted November 12, 2017. (Corresponding author: Sohini Sengupta.) S. Sengupta was with the Department of Electrical and Computer Engineering and the Department of Mathematics, University of Houston, Houston, TX 77204 USA. She is now with the Energous Corporation, San Jose, CA 95134 USA (e-mail: [email protected]). D. R. Jackson and S. A. Long are with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2783373

PRS LWA structures is given in [28] and [29]. The 2-D PRS type of LWA uses a PRS to confine the fields inside a radiating parallel-plate type of region, where the radiating wave is an outward propagating cylindrical wave that is fast with respect to free space, and hence radiates. If the PRS is periodic, there will be an infinite number of space harmonics (Floquet waves), but the fundamental space harmonic is the radiating one, and hence the 2-D LWA structure with a periodic PRS operates as a quasi-uniform type of LWA structure. Much less attention has been devoted to 2-D periodic LWAs, though some important works on 2-D periodic LWA structures of various forms have been presented [30]–[32]. A 2-D periodic LWA is defined here as one that has a 2-D periodic arrangement of elements on a surface (such as a grounded dielectric slab), and is excited with a simple source (such as a magnetic dipole) at the center. Fig. 1 shows the specific structure investigated here. The source excites a slow wave (slow with respect to free space) that radially propagates outward from the source. This wave, by itself, does not radiate. However, due to the periodicity, an infinite number of space harmonics (Floquet waves) are produced, and the structure is designed so that some of the space harmonics are fast waves, and hence radiate. The main purpose of this work is to study and characterize the nature of 2-D periodic LWAs for the first time. This paper has been motivated by the plasmonic phenomenon of directive beaming of light through a subwavelength aperture that is observed in the optics regime. In this case, an optical plane wave of light is incident on a subwavelength aperture that is surrounded by periodic corrugations on the exit face of a thin sheet of a plasmonic metal such as silver [33]–[36]. The field emerges on the exit side as a highly directive narrow beam of light. It has been shown that this phenomenon is due to leaky-wave radiation from a surface plasmon wave supported by the metal film [37], [38]. Although the motivation for the 2-D periodic structure is the optical directive-beaming phenomenon, the structure studied here is a microwave one. Instead of having a plasmon wave on a silver film as the slow wave that is launched from a source, a grounded dielectric slab is used, which supports the fundamental TM0 surface wave; this is used as the cylindrically propagating slow wave. However, the physics of the radiation from the space harmonics is the same, and hence an examination of the microwave structure also reveals the fundamental radiation physics in the plasmonic structure.

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In addition, the microwave structure may prove to be a useful antenna structure in the high microwave or millimeter wave bands, for applications that require a narrow elliptical beam using a very simple structure, where the beam is much wider in the H -plane compared to the E-plane (by roughly a factor of 10 for the case studied here). Preliminary work on a 2-D periodic LWA structure has been presented briefly in the conference papers [39]–[41], and this paper is a more detailed treatment of the work presented in [41]. This paper goes beyond [41], in that it provides an in-depth discussion of the current distribution on the patches, the optimization of the structure for broadside radiation, the mechanism for beam formation, and an understanding of the leaky modes from the field distribution. In addition to that, it further elucidates the dispersion behavior, grating lobes and radiation efficiency of the 2-D periodic LWA, which [41] briefly touched upon. This paper mainly focuses on the propagation aspects of the leaky modes on this type of structure, rather than the antenna characteristics, which will be explored further elsewhere. Since this paper is inspired by the plasmonic structure at optical frequencies, the results presented here provide insight into the mechanism of operation of the similar plasmonic structure as well. II. D ISCUSSION OF BASIC P RINCIPLES OF O PERATION The structure is shown in Fig. 1 and consists of a grounded dielectric substrate of thickness h. On the upper surface of the substrate there is a periodic arrangement of perfectly conducting patches with a uniform spacing in the x-direction given by a, and in the y-direction given by b. The dimensions a and b define the length and width of the unit cell. The narrow patches have length L (dimension along the x-direction) and width W (dimension along the y-direction). The structure is excited by a slot in the ground plane oriented along the y-direction that is modeled here as a magnetic dipole in the y-direction. It has been found that when the structure is optimized for maximum radiation at broadside, it produces a narrow and very directive beam at broadside that is much narrower in the E-plane (xz plane) than in the H -plane (yz plane). The magnetic dipole source launches a TM0 surface wave that propagates radially outward on the structure. If there were no metal patches, there would only be a pure TM0 surface wave. However, due to the presence of the patches, radiation from space harmonics occurs, and this forms the beam. In the E-plane radiation comes from the (−1, 0) space harmonic, and in the H -plane radiation comes from the (0, −1) space harmonic. In between these two planes the situation is more complicated, and this will be discussed later. The TM0 surface wave that is launched by the magnetic dipole when the patches are absent has an angular dependence of cosφ. The field of the surface wave by itself, in the absence of the patches, would have a magnetic vector potential in the air region that varies as (2)

A z = cos φ H1 (βTM0 ρ)e−αz0 z

(1)

where αz0 is the vertical decay constant, related to the phase constant of the surface wave βTM0 in the usual way [42]. It is noted that most of the energy in the surface wave is concentrated near the E-plane (φ = 0) due to the cosφ term, and this explains why the patches near the x-axis are the most important for the beamforming. An asymptotic expansion of the fields of the surface wave shows that the surface-wave field decays as 1/ρ 1/2 in the E-plane and as 1/ρ 3/2 in the H -plane. (Even though the cosφ term in (1) goes to zero in the H -plane, the fields are not exactly zero at φ = π/2, but are higherorder in nature, and decay algebraically faster.) Because the E-plane dimension is the most important, the periodic spacing a is the most important parameter in optimizing the structure for maximum radiation at broadside. Roughly speaking, the optimum periodic spacing a should be somewhat close to that predicted by ignoring the perturbation of the patches on the wavenumber of the leaky mode (i.e., using the wavenumber of the surface wave), and then requiring that the (−1, 0) Floquet wave radiate at broadside, which yields β−1,0 ≈ βTM0 −

2π = 0. a

(2)

The optimum spacing a ensures a main beam at broadside with maximum radiated power density at broadside. The size of the patches correlates with the attenuation constant of the leaky mode and hence with the beamwidth. The smaller the patches are, the narrower is the main beam (for patches that are much smaller than resonant length). Further details on the leaky modes and their role in the beam optimization are given in Section VI. For all of the results presented in this paper, the frequency of operation is 12 GHz, the substrate has a relative permittivity of 9.8 and a thickness of 1.27 mm, and is lossy unless stated otherwise. The only exception to this is for the truncated structure in Section IV, where simulation results for the truncated structure designed to operate at 24 GHz for a different substrate board have been presented. The dipole is located on the ground plane (z = −h), centered between two patches at x 0 = −a/2, y0 = 0. The design is always optimized by varying the unit cell dimension a, while maintaining the other dimension at b = a/1.2 (a somewhat arbitrary choice), in order to get the maximum power density radiated at broadside. With this method of optimization, we are able to get a very directive beam (main beam) at broadside. III. A NALYSIS M ETHODS FOR THE 2-D P ERIODIC S TRUCTURE For the theoretical analysis of the structure, we assume that the 2-D periodic leaky-wave antenna structure is infinite. For practical antenna fabrication purposes, the infinite structure can be truncated at the point where the current on the patches diminishes to the extent that it is negligible. All of the theoretical analysis for this structure is done using the periodic spectral domain immittance (SDI) method. The currents on the patches are modeled using basis functions that are sinusoidal along the length of the patch (in the x-direction) and vary as a “Maxwell function” along the width of the patch (in the y direction). Therefore, the surface current

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C. Leaky Mode Wavenumber for a Given Direction φ

density distribution on the patches is given by JsxP (x, y) =

N  n=1

an Bn (x, y) =

N 

an f n (x)g(y)

3

(3)

n=1

where an is the expansion coefficient of the nth order basis function Bn (x, y), and    L 1/π nπ x+ g(y) =  . (4) f n (x) = sin L 2 (W/2)2 − y 2 The functions fn (x) and g(y) give the variation of the basis function along x and y, respectively. The theoretical methods used for analyzing the 2-D periodic leaky-wave antenna are described briefly as follows. A. Reciprocity for the Far-Field Radiation Pattern To find the far-field radiation from the 2-D periodic leakywave antenna, reciprocity is used [7], [42]. This relates the far field of the structure produced by the magnetic dipole to the H y field at the dipole location inside the substrate due to a plane wave that is incident on the structure. The transverse equivalent network (TEN) [28], [43] transmission line model is used to model the layer of substrate with free space above and PEC ground below, to find the field inside the structure due to the plane wave itself. The periodic spectral domain method [7], [44] has been used to find the field inside the structure due to currents on the patches (which are periodic with a phase shift, due to the incident plane wave), with the current on the patches modeled as given in (3). The electric field integral equation (EFIE) [45], [46] is enforced on the central (0, 0) patch in the periodic problem to solve for the coefficients of the basis functions for the surface current density on the patches. B. Array Scanning Method to Find the Patch Currents The array scanning method (ASM) can be used for finding the near-field quantities such as the electric field or the current on an infinite periodic structure when it is excited by a single (nonperiodic) source [47], [48]. The ASM method can therefore be used to find the currents on the patches due to the magnetic dipole source in Fig. 1. In the ASM, the single magnetic dipole source is first replaced by an infinite phased array of identical magnetic dipole sources, having a set of arbitrary phasing wavenumbers (k x0 , k y0 ). Then, the current density on the patches in this periodic problem is calculated using the periodic spectral domain method. In this solution, the EFIE is enforced on the central (0, 0) patch to find the coefficients of the basis ∞ (x, y) functions, from which the surface current density, Jsx can be calculated for the structure with an infinite array of magnetic dipole sources. Finally, the surface current density on the patches for the single magnetic dipole, Jsx (x, y), is calculated as  π/b  π/a ab J ∞ (x, y, k x0 , k y0 )dk x0 dk y0. Jsx (x, y) = (2π)2 −π/b −π/a sx (5)

In the ASM described in Section III-B, the EFIE is enforced on the central patch. This leads to a system of linear equations to solve for the coefficients of the basis functions for the surface current. Setting the determinant of the coefficient matrix to zero yields a transcendental equation for the wavenumber of the propagating mode(s). The wavenumber kρLW of the fundamental (0,0) Floquet harmonic of the radially propagating leaky mode in the 2-D periodic LWA structure is a function of the azimuthal angle φ. The x- and y-components of the wavenumber kρLW are given by k x0 = kρLW (φ) cos φ k y0 = kρLW (φ) sin φ.

(6)

In solving for the wavenumber of the leaky wave, the value of the vertical wavenumber k z0 in the air region is given by 2 = k 2 −k 2 −k 2 for the ( p,q) Floquet mode/harmonic, where k z0 x y 0 the x- and y-components of the wavenumbers are k x = k x p = k x0 + 2π p/a and k y = k yq = k y0 +2πq/b, respectively. The choice of the square root

1/2 (7) k z0 = k02 − k x2 − k 2y is chosen so that the wave is “physical.” This means that it may sometimes be proper (exponentially decreasing vertically) or improper (exponentially increasing vertically), depending on the situation [49]. For any given Floquet wave, the wavenumbers are denoted as k x = β x − j αx k y = β y − j α y k t = k x xˆ + k y yˆ = β − j α

(8) (9)

where k x = k x p , k y = k yq . The following rules are then used to choose the proper/improper nature of each Floquet wave. 1) If β ·α = 0 (there is no attenuation, αx = α y = 0), then: a) if |β| < k0 (fast wave), βz > 0 b) if |β| > k0 (slow wave), αz > 0. 2) If β ·α > 0 (the wave is a forward wave in the transverse direction), then: a) if |β|< k 0 (fast wave), αz < 0 (improper) b) if |β| > k0 (slow wave), αz > 0 (proper). 3) If β · α < 0 (the wave is a backward wave in the transverse direction), then: a) if |β|< k 0 (fast wave), αz > 0 (proper) b) if |β| > k0 (slow wave), αz > 0 (proper). IV. R ADIATION PATTERN The radiation pattern is calculated numerically using the method described in Section III-A. A typical case is shown in Fig. 2 with the radiation patterns in the E-plane (φ = 0°), H -plane (φ = 90°), and the D-plane (φ = 45°). The substrate here is taken to be lossy, with a loss tangent of tanδ = 0.002. Here the length of the patches is L = 0.3 cm, the width is W = L/5, and the optimized dimensions of the unit cell are given by a = 2.2728 cm when b is chosen as b = a/1.2. (The choice of the b dimension is somewhat arbitrary.) For this case, the size of the patches is relatively small, which produces a more directive main beam with an enhancement factor

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Fig. 3. Radiation patterns in the H -plane, from reciprocity and the CAD formula at 12 GHz.

Fig. 1. Schematic of the 2-D periodic LWA, with the slot in the ground plane modeled as a magnetic dipole.

Fig. 4. Normalized radiation pattern (in dB) for the infinite structure at 24 GHz, calculated analytically using SDI and reciprocity.

Fig. 2. Radiation patterns in the E-plane, H -plane, and D-plane (φ = 45°) at 12 GHz.

of 110.89. The enhancement factor is defined as the magnitude of the far-field radiated at broadside from the structure divided by the magnitude of the far field at broadside from the same magnetic dipole source radiating in free space. The main beam at broadside is much narrower in the E-plane than in the H -plane. This is a fundamental characteristic of this type of antenna, and will be explained shortly in terms of the propagating leaky modes. The smaller the patches are, the narrower the main beam is, though it is always found that the beamwidth in the E-plane is roughly ten times smaller than the beamwidth in the H -plane. This is because smaller patches provide less disturbance to the propagating TM0 surface wave that propagates cylindrically outward from the dipole source, and hence the attenuation constants of the leaky modes are smaller when the patches are smaller (the leaky modes will be discussed later). Note that there are also narrow grating lobes present in the diagonal plane, which seems to be an unavoidable consequence of the 2-D periodic LWA structure. These grating lobes are not the usual grating lobes encountered in a phased array when the element spacing

is too large. More details about the origin of the grating lobes are given later. The E-plane pattern is determined by a leaky mode propagating in the E-plane direction, while the H -plane pattern is determined by a leaky mode propagating in the H -plane direction. This has been verified by plotting the E- and H -plane patterns obtained from a simple LWA CAD formula that is based on the wavenumber of a 1-D bidirectional leaky mode [50]. These patterns show excellent agreement with the exact patterns obtained by reciprocity in either plane, when the appropriate wavenumber for the relevant leaky mode is used in the CAD formula. A comparison for the H -plane is shown in Fig. 3. The agreement is seen to be good down to about −20 dB. The agreement for the E-plane pattern (omitted for brevity) is good down to about −30 dB. More discussion on the leaky modes is given in the following sections. A practical antenna has been designed at 24 GHz. The substrate height is 0.762 mm, the substrate relative permittivity is 4.5, and the substrate loss tangent is 0.002. The unit cell has dimensions a = 11.186 mm and b = a/1.2, and the patches have length L = 2.8 mm and width W = L/5. From reciprocity and SDI, we obtain the radiation pattern for the infinite structure as shown in Fig. 4. When a truncated structure [30.5 cm (12 in) along the x-axis, 22.9 cm (9 in) along the y-axis] is simulated in ANSYS Designer, we obtain the radiation pattern shown in Fig. 5. In ANSYS Designer, the source is realized by a loop of current close to the ground

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Fig. 5. Radiation pattern (directivity in dB) for the truncated structure in Fig. 4, from simulation in ANSYS Designer at 24 GHz.

plane and the substrate and ground are still infinite, while the patch array is finite (truncated). The directivity as measured in ANSYS Designer is 21.2 dB at broadside. Comparing these two patterns shows that a practical antenna can be realized, and the analytical calculations for the infinite structure agree well with the full-wave simulation for the truncated structure. V. C URRENT D ISTRIBUTION ON THE PATCHES The current at the center of the patches in the 2-D periodic leaky-wave antenna is calculated using the ASM discussed in Section III-B. In this case, the design parameters for the 2-D periodic leaky-wave antenna are as follows. The length of each patch is L = 0.4 cm, the width is W = L/5, and the dimensions of the unit cell are given by a = 2.41758 cm and b = a/1.2. The loss tangent of the substrate in this case is tanδ = 0.002. The period a has been optimized for maximum power density radiated at broadside. A larger patch length has been chosen here compared to Fig. 2 in Section IV (0.4 cm instead of 0.3 cm) because the leakage constants in both the E- and H-planes are then larger, and this allows for an examination of the leaky modes using full-wave simulations without expending excessive computational resources. This is because smaller attenuation constants require a larger structure to be simulated in order to see the same dynamic range of currents. The normalized current magnitude from ASM is shown in Figs. 6 and 7 for the E-plane (along the x-axis) and H -plane (along the y-axis), respectively. The black solid curve gives the asymptotic distribution of the patch currents based on the exponential decay of the leaky mode in each plane and an appropriate algebraic decay. For the E-plane, the leakywave field distribution over the surface of the dielectric is e−α E x /x 1/2 and for the H -plane it is e−α H y /y 3/2 , where α E and α H are the attenuation constants of the leaky mode in the E-plane (along the x-axis) and the H -plane (along the yaxis), respectively. For this case, the attenuation constants in the E-plane and H -plane are α E = 0.0121034k0 and α H = 0.082244k0 respectively, where k0 is the wavenumber in free space at this frequency. The current distribution obtained from simulation using the commercial full-wave simulator ANSYS Designer is also shown in both planes for comparison. In ANSYS Designer a fairly large array of patches (100 along x,

Fig. 6. Current distribution from ASM and ANSYS Designer on the patches in the E-plane.

80 along y) is simulated, and the source is realized by a loop of current close to the ground plane that emulates a magnetic dipole. The currents in each plane are normalized by forcing the curves to agree at one particular patch, chosen to be patch 9 and 22 for the E- and H -planes, respectively. This normalization removes a scaling factor that is due to the modeling of the magnetic dipole in ANSYS Designer and also the fact that the currents in ANSYS Designer were sampled at points that were at a fixed offset from the patch center. Along the surface of the dielectric, the leaky mode interferes with the space wave, and therefore the current distribution on the patches displays an interference pattern. In the absence of interference with the space wave, the current distribution from ASM would closely match the asymptotic (solid black) curve in Figs. 6 and 7. The current distribution obtained from both ANSYS Designer and ASM shows a similar interference pattern with the same beat period. The agreement is fairly good down to a level of about −30 dB. The ANSYS Designer simulation uses a finite structure, and thus there are some edge reflections that are partially responsible for the disagreement with the results from ASM, which assume an infinite structure. Such effects are expected for finite structures [51]. VI. O PTIMIZING P OWER D ENSITY AT B ROADSIDE Owing to the spatial periodicity of the 2-D periodic LWA structure, an infinite number of Floquet (space) harmonics will be excited on the structure. The wavenumber of the fundamental Floquet harmonic, as well as that of the higherorder Floquet harmonics of the leaky mode for a given azimuthal angle of propagation φ with the x-axis is calculated using the method described in Section III-C. Except for the (−1, 0) Floquet harmonic near the E-plane and the (0, −1) Floquet harmonic near the H -plane, which are fast waves and therefore radiating, all of the other harmonics (including the fundamental one) are slow waves and do not radiate. In the E-plane (along the x-axis), the (−1, 0) Floquet harmonic of the leaky mode produces the radiation and is the dominant radiating wave, whereas for the H -plane (along the y-axis), the

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Fig. 7. Current distribution from ASM and ANSYS Designer on the patches in the H -plane.

Fig. 8. Plot of normalized α and β along the x-axis, and the power density radiated at broadside, versus the period a.

(0, −1) Floquet harmonic of the leaky mode is the radiating wave. The normalized values of the phase constant β and attenuation constant α of the radiating waves [(−1, 0) for the E-plane and (0, −1) for the H -plane] are plotted along with the power density radiated in the far field at broadside with respect to a variation of the spacing between the patches a in the E-plane (along the x-axis) in Fig. 8, and the spacing between the patches b in the H -plane (along the y-axis) in Fig. 9. The attenuation constants and the phase constants are normalized by k0 . The following design parameters for the 2-D periodic leaky-wave antenna have been assumed: the substrate is lossless here, i.e., tanδ = 0, the length of each patch is L = 0.25 cm, the width is W = L/5, and the dimensions of the unit cell are given by a = 2.29705 cm and b = a/1.2. The period a has been optimized for maximum power density radiated at broadside. The values of a and b are then varied away from this optimum design. The substrate is chosen to be lossless here so that the attenuation constant of the leaky mode is only due to radiation and does not include material loss. This allows for the examination of the formation of a stopband, which theoretically only exists for a lossless structure. In Fig. 8, it can be observed that when the power radiated in the far field at broadside is maximum, the normalized phase

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Fig. 9. Plot of normalized α and β along the y-axis, and the power density radiated at broadside, versus the period b.

constant and the normalized attenuation constant of the leaky mode propagating along the x-axis have the same magnitude. Also, a region of open stopband can be observed where the phase constant is very low and decreases to zero while the attenuation constant has a much higher value than usual and then drops exactly to zero. This is precisely the same behavior that has been observed in 1-D periodic LWAs [52], [53]. In Fig. 9, it can again be observed that when the power radiated in the far-field region at broadside is maximum, the normalized phase constant and the normalized attenuation constant of the leaky mode propagating along the y-axis also have the same magnitude. This is the same property as the leaky mode along the x-axis observed in Fig. 8; but unlike the leaky mode along the x-axis, no open stopband region is observed for the leaky mode along the y-axis. VII. D ISPERSION B EHAVIOR OF L EAKY M ODES The wavenumber of the leaky mode that produces the radiation in the 2-D periodic leaky-wave antenna varies with the radial angle of propagation φ. We take the following design parameters for the 2-D periodic leaky-wave antenna: the substrate is lossless, i.e., tanδ = 0, the length of each patch is L = 0.25 cm, the width is W = L/5, and the dimensions of the unit cell are given by a = 2.29705 cm and b = a/1.2. This design has been optimized for maximum power density radiated at broadside. Once again the substrate is chosen to be lossless here so that the attenuation constant of the leaky mode is only due to radiation and does not include material loss. The normalized (normalized by k0 ) phase constant of the fundamental (0, 0) harmonic of the leaky mode is shown in Fig. 10 with respect to the angle φ. Similarly, the normalized attenuation constant of the fundamental harmonic of the leaky mode is shown in Fig. 11 with respect to the angle φ. Figs. 10 and 11 give the dispersion diagram for the entire infinite 2-D periodic LWA. Figs. 10 and 11 also indicate whether the radiating space harmonic is a proper or improper wave (i.e., exponentially decaying or increasing in the vertical z-direction.) More discussion of the proper/improper nature of leaky waves may be found in [50].

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Fig. 10. Dispersion diagram showing the variation of the phase constant with respect to the angle of propagation φ.

Fig. 12.

Distribution of the E-field for the E-plane mode.

Fig. 11. Dispersion diagram showing the variation of the attenuation constant with respect to the angle of propagation φ.

Fig. 13.

Distribution of the E-field for the H -plane mode.

From Figs. 10 and 11, we see that there are actually three separate leaky modes that are found. The one that is dominant along the x-axis, i.e., in the E-plane, is termed the E-plane mode or Mode E. The one that is dominant along the y-axis, i.e., in the H -plane, is termed the H -plane mode or Mode H. Then there is the third mode, Mode 3 that does not produce any radiation. The E-plane leaky mode produces the beam in the E-plane and the H -plane leaky mode produces the beam in the H -plane. Near the x-axis (within four degrees of the axis), the E-plane mode has a radiating (−1, 0) harmonic that is proper (denoted with a dashed line), as it is a slightly backward wave in the optimized structure, with a negative phase constant. On the other hand, the H -plane mode near the y-axis has a (0, −1) harmonic that is an improper wave (denoted with a solid line), as it is slightly forward with a positive phase constant. VIII. F IELD D ISTRIBUTION OF THE M ODES In order to understand the nature of the E-plane mode and the H -plane mode, we look at the distribution of the

7

E-field for these two modes. The distribution of the tangential E-field (in a viewing window perpendicular to the direction of propagation) for the E-plane mode and the H -plane mode is shown in Figs. 12 and 13, respectively. The field distribution for the E-plane mode viewed along the x-axis is very similar to that of the TM0 surface-wave mode, and therefore it is clearly a perturbation of the TM0 surface-wave mode, as expected. The phase constant of the fundamental (0,0) harmonic of the E-plane mode is close to that of the TM0 surface-wave mode. In the H -plane, along the y-axis, the dominant component of the TM0 surface-wave mode has a null [see (1)]. The higher-order field components of the surface-wave mode have a vertical electric field that is an odd function of x about an axis parallel to the y-axis through the dipole. The distribution for the vertical component of the E-field of the H -plane mode along the y-axis resembles a TE20 rectangular waveguide mode, with boundaries at x = ±a/2. The phase constant and attenuation constant of the H -plane mode are different from the phase constant and attenuation

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IX. G RATING L OBES

Fig. 14. Radiation pattern for φ = 50.53°, showing the main beam and the grating lobes. The grating lobe at θ = 38.09° that is predicted using a CAD formula is also shown.

Usually there are up to three grating lobes that can be observed in the radiation pattern off the principal planes. The origins of the narrow grating lobes that are observed in the radiation pattern in Fig. 2 can be explained and predicted using the fundamental principles of Floquet waves. We take the following design parameters for the 2-D periodic LWA: the substrate is lossless, i.e., tanδ = 0, the length of each patch is L = 0.25 cm, the width is W = L/5, and the dimensions of the unit cell are given by a = 2.29705 cm and b = a/1.2. This corresponds to the case used for the dispersion plots in Section VII. This period a has been optimized for maximum power density radiated at broadside. To illustrate, the E-plane leaky mode traveling radially along the angle φ00 = 50° is considered, and the (−1, −1) Floquet harmonic of the E-plane mode is considered, which has a wavenumber for the fundamental Floquet harmonic given as (in radians/m) LW = (1.0819− j 0.00004638)k0 = 272.11303− j 0.011664. k(0,0)

The phase constant vector of the (−1, −1) Floquet harmonic is at an angle in the xy pane of  LW β(0,0) sin φ00 − 2π/b −1 = 50.53°. (10) φ = tan LW cos φ − 2π/a β(0,0) 00 The magnitude of the phase constant vector of the (−1, −1) Floquet harmonic is (in radians/m)

LW

β

(−1,−1)   2  2 LW cos φ − 2π LW sin φ − 2π = + β β(0,0) 00 00 (0,0) a b = 155.1632. (11)

Fig. 15. beam.

Parts of the structure producing the grating lobes and the main

constant of the E-plane mode due to the very different field distributions. The field of the H -plane mode is formed mainly by a single pair of Floquet waves, namely the (−1, −1) and (1, −1) waves. These two Floquet waves, when added together with opposite amplitudes, form a field that has PEC boundary conditions at the edges of the unit cell in the x-direction, at x = ±a/2, and also at x = 0. These two Floquet modes of the H -plane mode become quite strong with opposite amplitudes once the structure has been optimized to radiate at broadside by choosing the optimum value of the period a. The vertical electric field thus has the form sin(2π x/a) inside the unit cell, resembling a TE20 rectangular waveguide mode. Evidently, this pair of Floquet modes becomes strongly resonant when the beam is scanned to broadside, since this corresponds to a cutoff condition for the corresponding TE20 waveguide. At cutoff, the magnetic dipole would launch a pair of waves bouncing back and forth in the x-direction between the waveguides walls, similar to the resonant pair of Floquet waves.

The attenuation constant of the (−1, −1) Floquet harmonic is taken as the attenuation constant of the fundamental (0, 0) harmonic projected onto the direction of propagation of the (−1, −1) Floquet harmonic. The complex wavenumber of the (−1, −1) Floquet harmonic in the direction φ = 50.53° is therefore LW = 155.1632 − j 0.011664 cos(50° − 50.53°). k(−1,−1)

(12)

The angle of the grating lobe in the plane φ = 50.53° is then





/k0 = 0.6649 rad = 38.09°. (13) θg = sin−1 β LW (−1,−1) This prediction works well since the actual grating lobes are at roughly around θ = 38.09° and 15° in the overall radiation pattern for the antenna in this plane. A simple CAD formula for the radiation pattern based on the complex wavenumber [50] is also very successful in predicting the location of the grating lobes as well, as can be seen from the radiation pattern for φ = 50.53° shown in Fig. 14. It has been observed that there are up to three grating lobes in the radiation pattern of the 2-D periodic leaky-wave antenna, depending on the φ angle. The grating lobes are very narrow in θ but are much broader in φ (the azimuthal angle). Two of the grating lobes in the radiation pattern are due to the (−1, 0) and (−1, −1) space harmonics of the E-plane leaky

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mode originating from the blue and purple shaded sectors (regions B and C) of the 2-D periodic LWA structure shown in Fig. 15. The third grating lobe is produced by the (0, −1) harmonic of the H -plane leaky mode originating from the part of the structure shaded in red (region D) in Fig. 15. These regions that produce the grating lobes as shown in the first and third quadrant of the structure in Fig. 15 are also mirrored in the second and fourth quadrants, but not shown for simplicity. The parts of the structure shaded in yellow in Fig. 15 produce the main beam from the (−1, 0) harmonics of the E-plane leaky mode near the x-axis (region A) and the (0, −1) harmonic of the H -plane leaky mode near the y-axis (region E). X. R ADIATION E FFICIENCY An approximate radiation efficiency for a leaky wave on a 1-D LWA (either uniform, quasi-uniform, or periodic) due to material loss can be calculated as αrad αrad er = = (14) αtotal αrad + αloss where αrad is the attenuation constant due to the radiation from the leaky wave and αloss is the attenuation constant due to the material loss. The attenuation term αloss is a sum of the dielectric loss (αd ) and the conductor loss (αc ). The conductor loss is a sum of the loss in the patches (α p ) and loss in the ground plane (αg ). This gives us the relation αloss = αd + α p + αg .

(15)

Equation (14) does not strictly apply to the 2-D periodic LWA that is studied here. For the 2-D periodic LWA there are two different leaky modes that play important roles in the radiation, and each one has a wavenumber that depends on the angle of propagation. However, the E-plane leaky mode is the dominant mode with the smallest attenuation constant. Therefore, an efficiency calculation that is based on this leaky mode is expected to give a worst case efficiency prediction. The attenuation constant of the surface wave is approximately equal to (αd + αg ). The attenuation constant due to radiation (αrad ) is calculated by making the dielectric and conductors completely lossless and calculating the attenuation constant of the leaky wave. The total attenuation constant αtotal (= αrad + αloss) is the attenuation constant of the leaky wave on a lossy structure. The loss in the patches α p can be calculated by modifying the determinantal equation for calculating the leaky-wave wavenumber (mentioned in Section III-C), to account for the loss due to the patches, by including a surface resistance Z s of the metal. Further details are omitted here. Results in Table I are shown for the following case for the E-plane leaky mode: the frequency is 12 GHz, the substrate has a relative permittivity of εr = 9.8, the substrate is lossy with a loss tangent tanδ = 0.002, and the thickness of the substrate is h = 1.27 mm. The length of each patch L is varied, and the width is W = L/5. The patches and ground plane are made of copper with a conductivity σ = 3×107 S/m. The design has been optimized for maximum power density radiated at broadside by adjusting the period a while keeping

9

TABLE I R ADIATION E FFICIENCY V ERSUS L ENGTH OF PATCHES

the width at b = a/1.2. The attenuation constant of the surface wave with a lossy copper ground plane is 0.00030942k0, where k0 is the wavenumber in free space. Table I gives the radiation efficiency for three different cases with different lengths of patches with the period a optimized to produce maximum power density at broadside. Table I also lists the attenuation constant of the E-plane leaky mode with and without considering the loss from the patches, and the attenuation constant due to the radiation from the E-plane leaky mode. From Table I, we see that the radiation efficiency increases as the length of the patches is increased. XI. C ONCLUSION A 2-D periodic leaky-wave antenna has been studied with attention to the behavior of the leaky modes and how they form the main beam and grating lobes in the pattern. This is the first time that an in-depth study of a 2-D periodic leaky-wave antenna has been presented. The structure examined consists of a periodic array of metal patches over a grounded dielectric slab, with the structure excited by a simple magnetic dipole in the ground plane in the middle of the structure. The dipole launches a radially propagating TM0 surface wave, which becomes leaky due to radiation from space harmonics (Floquet waves) of the propagating wave. This structure is motivated by a plasmonic directive-beaming structure, as the fundamental physics of beamforming is the same between the optical plasmonic structure and the microwave structure examined here. The structure presented may also find application as a simple type of antenna suitable for mm-wave applications that require elliptical-shaped beams. The investigation revealed that two leaky modes are responsible for the formation of the main beam, called here the E-plane mode and the H -plane mode. The E-plane mode is responsible for the E-plane pattern of the main beam via radiation from the (−1, 0) space harmonic (Floquet wave). The H -plane mode is responsible for the H -plane pattern via radiation from the (0, −1) space harmonic. The E-plane mode is a perturbation of the TM0 surface-wave mode, which gets perturbed by the patches. The H -plane mode is a different type of mode, having a field configuration that more closely resembles that of the TE20 rectangular waveguide mode along the y-axis. The attenuation constant of the H -plane leaky mode is much higher than that of the E-plane leaky mode, and consequently the beamwidth of the main beam is much narrower in the E-plane than in the H -plane, with a typical ratio of 10:1 for the beamwidths. Numerical calculations of

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the currents on the patches have verified that the current distribution matches well with that predicted by the leaky modes, though interference is observed between the leaky modes and a space-wave type of field. This verifies the leakywave principle of operation for this structure. When the lossless 2-D periodic LWA structure is optimized to get maximum power density radiated at broadside, the condition |β| = α is fulfilled for the radiating harmonics of both the E-plane leaky mode and the H -plane leaky mode, consistent with 1-D leaky-wave antenna theory. Grating lobes are produced by the (−1, 0) and (−1, −1) space harmonics of the E-plane mode and the (0, −1) space harmonic of the H -plane mode. These grating lobes are evidently always present off of the principal planes in this type of structure, once the structure has been optimized for broadside radiation. The grating lobes are very narrow, however, and may not be a serious issue for truncated finite-size structures (this remains to be investigated). The radiation efficiency is found to increase as the size of patches is increased, since the leakage rate increases relative to the material attenuation. R EFERENCES [1] G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. 4, no. 4, pp. 666–671, Oct. 1956. [2] N. Alexopoulos and D. R. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-32, no. 8, pp. 807–816, Aug. 1984. [3] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 9, pp. 976–987, Sep. 1985. [4] D. R. Jackson and A. A. Oliner, “A leaky-wave analysis of the highgain printed antenna configuration,” IEEE Trans. Antennas Propag., vol. AP-36, no. 7, pp. 905–910, Jul. 1988. [5] D. R. Jackson, A. A. Oliner, and A. Ip, “Leaky-wave propagation and radiation for a narrow-beam multiple-layer dielectric structure,” IEEE Trans. Antennas Propag., vol. 41, no. 3, pp. 344–348, Mar. 1993. [6] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” Proc. Inst. Electr. Eng.–Microw., Antennas Propag., vol. 148, no. 6, pp. 345–350, Dec. 2001. [7] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas—Part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [8] C. Mateo-Segura, M. García-Vigueras, G. Goussetis, A. P. Feresidis, and J. L. Gómez-Tornero, “A simple technique for the dispersion analysis of Fabry-Pérot cavity leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 60, no. 2, pp. 803–810, Feb. 2012. [9] L. Moustafa and B. Jecko, “EBG Structure with wide defect band for broadband cavity antenna applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 693–696, 2008. [10] C. Mateo-Segura, A. P. Feresidis, and G. Goussetis, “Broadband leakywave antennas with double-layer PRS: Analysis and design,” in Proc. Eur. Conf. Antennas Propag. (EuCAP), Apr. 2011. [11] Y.-F. Lu and Y.-C. Lin, “Design and implementation of broadband partially reflective surface antenna,” in Proc. IEEE Int. Symp. Antennas Propag., Spokane, WA, USA, Jul. 2011, pp. 2250–2253. [12] Y. Ge, K. P. Esselle, and T. S. Bird, “The use of simple thin partially reflective surfaces with positive reflection phase gradients to design wideband, low-profile EBG resonator antennas,” IEEE Trans. Antennas Propag., vol. 60, no. 2, pp. 743–750, Feb. 2012. [13] D. Kim, J. Ju, and J. Choi, “A mobile communication base station antenna using a genetic algorithm based Fabry-Pérot resonance optimization,” IEEE Trans. Antennas Propag., vol. 60, no. 2, pp. 1053–1058, Feb. 2012. [14] A. Hosseini, A. T. Almutawa, F. Capolino, and D. R. Jackson, “V-band Wideband Fabry-Pérot cavity antenna made of thick partially reflective surface,” in Proc. IEEE Int. Symp. Antennas Propag., Fajardo, Puerto Rico, Jun. 2016, pp. 349–350.

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[37] D. R. Jackson, A. A. Oliner, T. Zhao, and J. T. Williams, “Beaming of light at broadside through a subwavelength hole: Leaky wave model and open stopband effect,” Radio Sci., vol. 40, no. RS6S10, pp. 1–12, Sep. 2005. [38] D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Exp., vol. 16, no. 26, pp. 21271–21281, Dec. 2008. [39] S. Sengupta, D. R. Jackson, and S. A. Long, “Properties of microwave and optical 2-D periodic leaky wave antennas,” in Proc. Texas Symp. Wireless Microw. Circuits Syst., Apr. 2015, pp. 1–4. [40] S. Sengupta, D. R. Jackson, and S. A. Long, “Examination of radiation from 2D periodic leaky-wave antennas,” in Proc. USNC-URSI Radio Sci. Meeting, Jul. 2014, p. 76. [41] S. Sengupta, D. R. Jackson, and S. A. Long, “Propagation characteristics of leaky waves on a 2D periodic leaky-wave antenna,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 413–415. [42] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Hoboken, NJ, USA: Wiley, 2001. [43] L. B. Felsen and M. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ, USA: Prentice-Hall, 1973. [44] T. Itoh and W. Menzel, “A full-wave analysis method for open microstrip structures,” IEEE Trans. Antennas Propag., vol. AP-29, no. 1, pp. 63–68, Jan. 1981. [45] T. K. Sarkar, “A note on the choice of weighting functions in the method of moments,” IEEE Trans. Antennas Propag., vol. AP-33, no. 4, pp. 436–441, Apr. 1985. [46] T. K. Sarkar, A. R. Djordjevic, and B. M. Kolundzija, “Method of moments applied to antennas,” in Handbook of Antennas in Wireless Communications. Boca Raton, FL, USA: CRC, 2001, ch. 8. [47] R. A. Sigelmann and A. Ishimaru, “Radiation from periodic structures excited by an aperiodic source,” IEEE Trans. Antennas Propag., vol. AP-13, no. 3, pp. 354–364, May 1965. [48] F. Capolino, D. R. Jackson, D. R. Wilton, and L. B. Felsen, “Comparison of methods for calculating the field excited by a dipole near a 2-D periodic material,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1644–1655, Jun. 2007. [49] T. Zhao, “Analysis and design of 2-D periodic leaky wave antennas using metal patches or slots,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Houston, Houston, TX, USA, Aug. 2003. [50] C. Caloz, D. R. Jackson, and T. Itoh, “Frontiers in antennas: Next generation design and engineering,” Leaky-Wave Antennas, F. B. Gross, Ed., 1st ed. New York, NY, USA: McGraw-Hill, 2011, ch. 9. [51] A. Foroozesh and L. Shafai, “2-D truncated periodic leaky-wave antennas with reactive impedance surface ground planes,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 2006, pp. 15–18. [52] P. Burghignoli, G. Lovat, and D. R. Jackson, “Analysis and optimization of leaky-wave radiation at broadside from a class of 1-D periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2593–2604, Sep. 2006. [53] S. Otto, A. Rennings, K. Solbach, and C. Caloz, “Transmission line modeling and asymptotic formulas for periodic leaky-wave antennas scanning through broadside,” IEEE Trans. Antennas Propag., vol. 59, no. 10, pp. 3695–3709, Oct. 2011.

Sohini Sengupta was born in Kolkata, India, in 1985. She received the bachelor’s degree in electronics and communication engineering from the Heritage Institute of Technology, Kolkata, in 2007, and the M.S. and Ph.D. degrees in electrical engineering from the University of Houston, Houston, TX, USA, in 2012 and 2016, respectively. From 2011 to 2016, she was a Teaching Assistant with the Department of Electrical and Computer Engineering, University of Houston, where she was a Post-Doctoral Fellow and a Lecturer from 2016 to 2017. She has been an Antenna Design Engineer with Energous Corporation, San Jose, CA, USA, since 2017. She has authored a journal paper on linear microstrip series-fed antenna arrays and several conference papers on different areas. Her current research interests include microstrip antennas, arrays, numerical methods, leaky-wave antennas, radar cross-sectional reduction techniques, and antenna miniaturization techniques.

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David R. Jackson (F’99) was born in St. Louis, MO, USA, in 1957. He received the B.S.E.E. and M.S.E.E. degrees from the University of Missouri, Columbia, MO, USA, in 1979 and 1981, respectively, and the Ph.D. degree in electrical engineering from the University of California, Los Angeles, CA, USA, in 1985. From 1985 to 1991, he was an Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX, USA, where he was an Associate Professor from 1991 to 1998, and has been a Professor since 1998. His current research interests include microstrip antennas and circuits, leaky-wave antennas, leakage and radiation effects in microwave integrated circuits, periodic structures, and electromagnetic compatibility and interference. Dr. Jackson currently serves as the Chair of the U.S. National Committee of URSI (USNC-URSI), the International Union of Radio Science. He has been the Chair of the Distinguished Lecturer Committee of the IEEE Antennas and Propagation Society (AP-S), the Chair of the Transnational Committee of the IEEE AP-S, the Chair of the Chapter Activities Committee of the AP-S, a Distinguished Lecturer for the IEEE AP-S, a Member of the AdCom for the AP-S, and an Associate Editor of the IEEE T RANSAC TIONS ON A NTENNAS AND P ROPAGATION . He served as the Chair of the MTT-15 (Microwave Field Theory) Technical Committee, the Chair and the Secretary of Commission B of USNC-URSI, and also as an Associate Editor for the Journal Radio Science and the International Journal of RF and Microwave Computer-Aided Engineering. He is on the Education Committee of the AP-S and on the MTT-15 (Microwave Field Theory) Technical Committee of the IEEE Microwave Theory and Techniques Society.

Stuart A. Long (LF’11) was born in Philadelphia, PA, USA, in 1945. He received the B.A. (magna cum laude) and M.E.E. degrees in electrical engineering from Rice University, Houston, TX, USA, in 1967 and 1968, respectively, and the Ph.D. degree in applied physics from Harvard University, Cambridge, MA, USA, in 1974. He was a faculty member with the Department of Electrical and Computer Engineering, University of Houston, for the past 44 years. At the University of Houston, he is currently an Associate Dean of Undergraduate Research and the Honors College. He was the Chair of the Department of Electrical and Computer Engineering from 1981 to 1995 and the Interim Dean of the Honors College from 2008 to 2009. His current research interests include the broad area of applied electromagnetics and more specifically in microstrip and dielectric resonator antennas. Dr. Long is a member of Phi Beta Kappa, Tau Beta Pi, Sigma Xi, and Commission B of URSI. He was elected to membership in the Electromagnetics Academy in 1990. He was a recipient of the IEEE Millennium Medal in 2000, the IEEE Antennas and Propagation Society Outstanding Service Award in 2007, and the IEEE Antennas and Propagation Society (IEEE AP-S) John Kraus Antenna Award in 2014. He was the first recipient of the University of Houston Career Teaching Excellence Award in 2009. In 2010, he received the Esther Farfel Award and the Highest Faculty Award from the University of Houston. He served as an IEEE AP-S Distinguished Lecturer from 1992 to 1994. He was an Elected President of the IEEE AP-S in 1996. He was the TAB Magazines Chair and a member of the Periodicals Review Committee from 1997 to 1999, and a Member-at-Large of the IEEE Publications Activities Board from 1998 to 2003. He served on the IEEE Technical Activities Board, the Spectrum Editorial Board from 2002 to 2005, the Interim Vice Chancellor and Vice President for Research and Technology Transfer, from 2010 to 2011. He was elected to serve on the Board of Directors of the IEEE for 2005 to 2006, as a Director of Division IV.

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A Scalable Multiharmonic Surface-Potential Model of AlGaN/GaN HEMTs Qingzhi Wu, Yuehang Xu , Senior Member, IEEE, Yongbo Chen, Yan Wang, Wenli Fu, Bo Yan, and Ruimin Xu, Member, IEEE Abstract— An accurate physical model for GaN high-electronmobility-transistors (HEMTs) device is imperative and crucial for circuit design and technology optimization. In this paper, a scalable large-signal surface-potential (SP) model of AlGaN/GaN HEMTs is presented. The drain current model is capable of accurately modeling the self-heating effect and trapping effect. The self-heating effect is modeled by embedding temperature increment into free-carrier mobility model, and the trapping effect is modeled by introducing an indirect variable effective gate voltage Vgseff . Moreover, the scaling and multiharmonic characteristics of the SP model are studied for the first time. The geometry-dependent thermal resistance Rths is identified by the electrothermal finite-element method simulations, which is scalable with the gate width W and power dissipations Pdiss of the device. Single-tone on-wafer load–pull measurements at operating frequency 8 GHz is carried out for verification purpose. Accurate predictions of the static (dc) I−V , pulsed-gate-and-drain I − V , S-parameters up to 40 GHz and large-signal harmonic performance (the fundamental, second- and third-harmonics output power, and power-added efficiency) for the devices with different gate peripheries have been achieved by the proposed model. The results of this paper can pave the way for the full application of the physical-based model in circuits design. Index Terms— AlGaN/GaN high-electron-mobility transistors (HEMT), scalable, self-heating, surface potential (SP), trapping effect.

I. I NTRODUCTION

T

HE outstanding intrinsic properties of GaN material such as high-electron mobility, high sheet carrier density in excess of 1013 cm−2 , and high breakdown field, make it a promising candidate to replace silicon and silicon carbide in power devices [1], [2]. Therefore, based on the excellent material properties of GaN, the AlGaN/GaN high-electronmobility transistors (HEMTs) with remarkable outstanding

Manuscript received July 5, 2017; revised October 17, 2017; accepted November 5, 2017. Date of publication December 7, 2017; date of current version March 5, 2018. This work was supported in part by the National Natural Science Foundation of China under Grant 61474020, in part by the China Postdoctoral Science Foundation under Grant 2016T90844, and in part by the National Key Project of Science and Technology. (Corresponding author: Yuehang Xu.) Q. Wu, Y. Xu, B. Yan, and R. Xu are with the EHF Key Laboratory of Fundamental Science, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). Y. Chen is with the Chengdu Hiwafer Co. Ltd., Chengdu 611731, China. Y. Wang is with the Institute of Microelectronics, Tsinghua University, Beijing 100084, China. W. Fu is with the National Key Laboratory of Science and Technology on Space Microwave, China Academy of Space Technology, Xi’an 710100, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2775639

performance in high-power and high-frequency applications in solid-state power microwave and millimeter-wave devices have been widely researched [3]–[6]. However, the existence of self-heating effect and charge-trapping effect limits the wide application of GaN devices and increases the difficulty of modeling [7]. Moreover, it is essential to ensure the effectiveness of physical model in different gate peripheries such as the gate width W . Therefore, an accurate compact physical model with scalability and physical describing ability is urgently demanded. In recent years, AlGaN/GaN HEMTs models based on the surface-potential (SP) theory have been widely attracted attention [8]–[12]. The self-heating effect is taken into the model mainly through the methods such as embedding the temperature term directly into SP calculation process [10], taking the temperature increment into carrier mobility expression [11], and incorporating both T1 (caused by power dissipation) and T2 (caused by heat diffusion from channel through substrate and backside) into temperature modeling to update drain current expression [12], which has shown to be accurate. However, scalable characteristic for different gate geometrical parameters into the SP model has been reported yet, which is essential for establishing an integrated and accurate physical model with exact physical interpretation. Furthermore, to acquire excellent performance of power amplifier in circuit design, the large-signal model is expected to accurately predict high-order harmonic productions under different impedance terminations. Previously, the drain current and gate capacitance models based on SP theory have been developed [13] and characterization method of self-heating and trapping effects is described based on the proposed model [14], which is the foundation for the subsequent development of scalable multiharmonic largesignal model. Therefore, in this paper, a scalable multiharmonic SP model is developed which integrates self-heating and charge-trapping effects through temperature-dependent carrier mobility model and effective gate–source voltage, respectively. Moreover, to obtain accurate model scalable ability, the nonlinear thermal resistance is modeled as the function of the geometrical parameters such as gate width W and the power dissipation Pdiss by using finite-element method (FEM) simulation method (ANSYS v.15.0), which is more convenient than the pulsed I−V measurement method [15]. This paper is organized as follows. In Section II, an improved Ids model including self-heating and charge-trapping effect suitable for different gate peripheries is described in detail. The specific extraction process for geometrical para-

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TABLE I E XTRACTED PARASITIC PARAMETERS FOR 4 × 100 μm A L G A N/G A N HEMT

Fig. 1. Schematic cross-sectional structure of a general Alx Ga1−x N/GaN HEMT and related conduction band diagram for zero gate bias depicting the 2-DEG formation, with gate length L, gate width W , and Fermi potential E f .

Fig. 2. Large-signal model topology for AlGaN/GaN HEMTs. The red-box parts represent SP-based drain current and gate capacitors models.

meters dependent thermal resistance Rths based on the FEM is explained, specifically to obtain channel temperature increment T . The trapping effect model is described briefly. Then the dc-I −V and pulsed I −V characteristics are discussed. In Section III, for verification, the comparisons between modeled and measured results including small-signal S-parameters, the fundamental, second- and third-harmonics output power, power-added efficiency, and gain are demonstrated. Finally, Section IV is the conclusion. II. M ODEL D ESCRIPTION The cross-sectional conceptual view of the Alx Ga1−x N/GaN HEMT device considered in this paper is shown in Fig. 1. SP φs is defined as the potential of the bottom of the conduction band at the AlGaN/GaN interface relative to ground. In the vicinity of interface in GaN layer, a potential well is formed because of the bandgap difference between AlGaN and GaN layers. Then sheet electron charges are confined in the potential well to form 2-DEG due to the spontaneous and piezoelectric polarization induced by lattice mismatch between AlGaN and GaN. Fig. 2 shows the topology of the proposed scalable largesignal model. The most important intrinsic elements mainly include the bias dependent gate–source capacitor Cgs , gate– drain capacitor Cgd , and nonlinear drain–source current Ids which are developed based on the analytic SP solution. The linear extrinsic elements consist of the parasitic resistances,

Fig. 3. Modeling flowchart for the scalable large-signal model of AlGaN/GaN HEMTs.

inductances, and capacitances which follow the linear scaling principle [16]. The parasitic elements are first extracted through cold FET [17], [18] S-parameters by using the extraction method in [19] and numerical optimization from 0.1 to 40 GHz. A GaN HEMT with gate length 0.25 μm and gate width 4×100 μm is first used in this paper. The extracted parasitic parameters are shown in Table I. The most important part in the modeling process is the establishment of nonlinear drain current and gate capacitances based on the SP. Fig. 3 demonstrates the complete modeling flowchart with analytic fermi level E f solution as the first step. Then the nonlinear drain current with charge-trapping effect and gate capacitances is obtained based on the SP. The self-heating effect is described through taking channel temperature increment T into temperature-dependent carrier mobility model. The geometry-dependent thermal resistance Rths is identified by the electrothermal FEM simulations. Following the large-signal model through embedding core models into the ADS symbolically defined devices (SDD), the final complete scalable large-signal model is developed by using suitable scaling rules. A. Modeling of Drain Current Based on SP The Ids model plays an important role in the modeling process which greatly affects the stability and output

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performance of the device. The sheet-charge approximation is considered to identify the diffusion and drift currents. The drain current is equal to the sum of above two kinds of currents with the current density expressed as Jn = Jdrift + Jdiff = qn s μeo E n + q Dn ∇n s

TABLE II PARAMETER VALUES U SED IN THE M ODEL

(1)

where E n is channel electric field along with the x-direction indicated as E n = −∇φs . μeo is effective carrier mobility. Dn is diffusion coefficient denoted by Einstein relation, reflecting the difficulty degree of carrier movement when existing concentration gradient. Under the assumption of the gradual channel approximation, the drain current in our previous work can be written as [13] Ids =

μe (T )c0 W (v gt + v T − φsm )(φsd − φss ) δ L × (1 + λVds + λb · ebk (Vdg −Vbr ) )

Vth = ϕ B (x Al) − E C (x Al ) −

(2)

q N D dd2 qσ (x Al) − (dd + di ) 2ε(x Al ) ε(x Al ) (3)

where W and L are gate width and gate length. c0 = (x Al )/d with (x Al ) being dielectric constant and d being thickness of Alx Ga1−x N layer and Al mole content x Al . v gt = Mtr × Vgs − Vth with gate–source voltage Vgs and transconductance modulation factor Mtr . Vth is the threshold voltage expressed as (3) with schottky barrier height ϕ B , doping concentration of n-AlGaN layer ND , polarization-induced charge density at the interface σ , and the conduction band offset at the AlGaN/GaN interface E C . v T = kT/q is the thermal voltage. φsm = (φss + φsd )/2 is the average SP with φss and φsd calculated as the SP at source and drain sides, respectively. The saturation of carrier velocity because of a decrease in mobility when the lateral electric field increases is denoted by δ = 1/[1 + 2 − φ 2 )]−1/2 . ρ = μ /ν , ν is the saturation velocity. ρ 2 (φsd eo s s ss λ is the channel length modulation effect parameter. And the term λb ×ebk (Vdg −Vbr ) describes the breakdown characteristic of device. With the increasing power dissipation caused by higher transistor operating voltage and current, the self-heating effect is exacerbated strongly in GaN HEMT. Therefore, accurate characterization of self-heating effect is critical to establishing explicit current model. Here, the self-heating effect is modeled by embedding the temperature increment T caused by dissipated power into the carrier mobility model [11], expressed as    1 1 − (4) μe (T ) ≈ μeo 1 + PT T T0 where T = T + T0 (T0 = 300 K) with temperature increment T , the function of Pdiss described as T = Rth × Pdiss and Pdiss = Ids ×Vds accounts for the static and quasi-static intrinsic power dissipation. Rth is the thermal resistance of the device which is obtained through the 3-D finite-element simulation in ANSYS software by building the structural model. PT is set as 5.3 × 102 K for a good approximation.

The μeo is the effective carrier mobility expressed as   μeo = μl / 1 + m 1 E V + m 2 E V2

(5)

which is dependent on the charge density in the channel relevant to the vertical field in GaN layer. μl is the low-field mobility, m 1 and m 2 are fitting parameters to be extracted from experimental data to model the vertical field dependence of carrier mobility. E V is the effective vertical electric field indicated as E V = ε(v gt − φsm )/dεGaN . After updating the improved carrier mobility model μe (T ), the new drain current expression is   PT Ids Vds Rth (6) Ids = Ids0 1 − TT0 where Ids0 is the calculated drain current without selfheating effect through μeo . The equation is developed as a quadratic variable with Ids as the independent variable. Therefore, the final drain current expression including selfheating effect is √ −b  ± b2 − 4a  c Ids = (7) 2a  where a  = Vds Rth , b = T0 + (PT /T0 )Ids0 Vds Rth − Ids0 Vds Rth , c = −T0 Ids0 . The definitions and values of model parameters for 4 × 100 μm reference device are given in Table II. B. Charge/Capacitance Models The assigning principle of channel charge presented by Ward [20] is followed to obtain the gate charge Q g written

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as:

 Qg = −

L

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q W n s (Vgs , Vch )d x

0

 =−

L

W c0 [v gt − φs (x)]d x  2  WLc0 v gt − v gt (φss + φsd − v T ) − v T φsm =− (8) 2 + φ2 + φ φ ) +(1/3)(φss θ ss sd sd 0

where θ = v gt − φsm + v T . The drain charge Q d is obtained by integrating x from source to drain. We obtain Q d = −(W Lc0 /120θ 2 )   3 + 8φ 3 + A φ 2 + A φ 2 12φsd sd sd ss ss ss × +Bsd φsd + Bss φss + C1 φsd φss + C2

(9)

Fig. 4. Thermal conduction in AlGaN/GaN HEMT device caused by selfheating effect.

where Asd = 24φss − 15(3 v T + 4 v gt ), Ass = 16φsd − 5(5v T + 8 v gt ), Bsd = 20(v gt + v T )(5 v gt + 2v T ), Bss = 20(v gt + v T )(4v gt + 2v T ), C1 = −10(5v T + 8v gt ), and C2 = −60v gt (v gt + v T )2 . Then the source charge Q s = −Q g − Q d can be obtained according to the charge conservation. Therefore the capacitance is obtained as Ci j = ki j ( Q i / V j ), ki j = 1(−1) when i = j (i = j ), where i and j represent the drain, gate, and source terminals.

e

e

C. Model Scalability Characterization The characterization of scalability is an indispensable step in the development process of a compact model. Up to now, scalable models of multiple fingers or multiple cells devices have been widely investigated [21]–[23]. By performing the full-wave electromagnetic analysis, the accurate parasitic effects are obtained of the actual device layout (manifold, air bridge, and via holes) [24], [25]. The simple linear scaling rules are employed for extrinsic equivalent circuit elements of different geometries, which show good accuracy. However, for the intrinsic drain current model, the linear scaling rule could be inapplicable due to the self-heating effect is significant for the AlGaN/GaN HEMTs device and the thermal resistances of the device demonstrate to be geometry dependent. Compared with the traditional empirical model, the SP model itself has a scaling capacity for the gate peripheries, expressed as the W /L term in drain current expression. In order to accurately characterize the self-heating effect at different gate geometries, the thermal resistance is expressed as a function of the gate peripheries by establishing a structural model. Due to the convenience for extract thermal parameters, the electrothermal FEM simulation is widely used. Moreover, compared with the 2-D simulation, the 3-D structural model has better accuracy because the 3-D effects are fully considered [26]. In addition, the simulations are very time-saving and suitable for different-in-size devices to determine the thermal parameters. Therefore, in this paper, the 3-D simulation process in ANSYS (v.15.0) software is carried out to extract the geometry-dependent thermal resistance. The developed 3-D thermal simulation for AlGaN/GaN with four-gate fingers is structured as this: separations of the gate–source and gate– drain are 1 and 2 μm, respectively. The each gate finger width Wg is 100 μm and the gate length is 0.25 μm. In order to simplify the FEM simulation, only the main layers including barrier layer, buffer layer, and substrate are investigated, since

Fig. 5. Simulation results of Rth for different AlGaN/GaN HEMTs. (a) Rth versus the power dissipations Pdiss with multiple gate width. (b) Rth versus gate width W under various dissipated power.

other layers (e.g., space layer) are considered too thin to affect thermal conduction within the device. The thickness of SiC substrate, GaN buffer layer, and AlGaN barrier layer is set as 100 μm, 1.5 μm, and 20 nm, respectively. After setting the basic structural parameters, the steadystate analysis results of heat distribution due to self-heating are shown in Fig. 4. It is obvious that power dissipation in the device causes the self-heating effect at the channel of the device and thermal profile can be approximated as elliptical shape which is consistent with the actual situation. Therefore, the relationship between Rth and dissipated power Pdiss is obtained as Fig. 5(a), It shows that the Rth increases with the Pdiss , which can be attributed to the nonlinear thermal conductivity of the device with respect to the temperature [27]. In order to obtain the scalability of the Rth with device periphery, different unit gate finger width Wg and number Ng have been involved to be investigated. Here, we define Wsc and Nsc as the scaling factor for the unit gate width Wg and number Ng , expressed as Wsc = Wg /Wgre

(10)

Nsc = Ng /Ngre Wg = W/Ng

(11) (12)

where Wgre and Ngre is the unit gate width and gate finger number of the reference device and we select Wgre = 100 μm and Ngre = 4 as a reference. W is the total gate width of device. The gate geometry of devices studied in this paper is 4 × 50, 4 × 100, and 6 × 100 μm. Fig. 5(b) shows the Rth versus W under the multiple Pdiss . It can be seen that with the increase of W , Rth exhibits a decreasing trend and finally achieves stability, which is consistent with the heat transfer mechanism [23]. Then, the scalable

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Fig. 6. Measured (circles) and simulated (lines) I−V characteristics for 4 × 100 μm AlGaN/GaN HEMT (a) without and (b) with SHE for Vds = 0 to 35 V with step 1 V and Vgs = −4 to 0 V, 0.2-V step from bottom to up.

thermal resistance Rths versus Wsc and Nsc in conjunction with the Pdiss is modeled as the expression Rth (Pdiss ) ∗ F(Wsc ) NSC 3 i Rth (Pdiss ) = kpi Pdiss , i = 0, 1, 2, 3 Rths =

F(Wsc ) =

i=0 3

λwj Wscj ,

j = 0, 1, 2, 3.

(13) (14)

Fig. 7. Measured (circles) and simulated (lines) dc–I−V for the differentin-size AlGaN/GaN HEMTs devices. (a) 4 × 50 μm. (b) 4 × 100 μm. (c) 6 × 100 μm.

(15)

j =0

The Rths denotes the thermal resistance after scaling with the device geometries. Rth is modeled as a function of Pdiss and F(Wsc ) is the function of scaling factor Wsc . kpi and λwj are fitting parameters. The effectiveness of the temperature-dependent carrier mobility model accounting for the self-heating effect is illustrated in Fig. 6. The developed current–voltage model with self-heating effect are shown in Fig. 6(b) with taking pulsedgate-and-drain I−V (PIV) measurement Fig. 6(a) at Vgsq = 0 V and Vdsq = 0 V as reference due to the self-heating and trapping effects are negligible in this case. A significant selfheating effect can be observed in high-current regimes on account of the electronic mobility and saturation velocity dependence on temperature in the channel. Although the modeling method with take temperature increment into mobility is not so comprehensive, possessing the advantage of fewer parameter and easy to implement. The excellent agreement between modeled and measured experimental results confirms the effectiveness of the model to predict AlGaN/GaN power HEMT dc behavior. Fig. 7 shows the measured and simulated dc-I−V behavior of the device with different gate width. Three devices are all biased at Vgs = −4 to 0 V, with 0.2-V steps from bottom to top, Vds = 0 to 35 V, with 1-V step. It is clearly seen that the scaled model can give a perfect prediction of the Ids for different size devices, and the significant self-heating effect is also accurately characterized by the improved nonlinear thermal resistances Rths . D. Modeling of Charge-Trapping Effect The charge-trapping effects owing to imperfections in fabrication process will lead to the gate-lag and drain-lag phenomena. At present, two approaches are generally performed

to characterize the traps-induced dispersion effect: analytical and empirically equivalent circuit methods [28], [29]. In this section, the effect of surface and substrate charge traps is characterized by pulsed I −V measurements from different static bias points using the Auriga/Au4850 pulsed dynamic I−V test systems with a pulsewidth (350 ns) and low duty cycle (0.1%). Thus self-heating effect can be avoided and the surface traps and substrate traps can be separated. The dynamic charge-trapping effect is ignored approximately [30]. The effect of charge traps on Ids can be seen as the impact on gate–source bias Vgs indirectly. The analytic expression of effective gate–source voltage is expressed as Vgseff = Vgs + ksurf(Vgsq − Vgspinchoff)(Vgs − Vgspinchoff) (16) + ksubs(Vdsq + Vdssub0)(Vds − Vdsq) where Vgsq and Vdsq are gate quiescent bias and drain quiescent bias, respectively. Coefficients ksurf and Vgspinchoff describe the surface-trapping effect caused by Vgs and Vgsq with respect to the pinch-off voltage. Coefficients ksubs and Vdssub0 are used to characterize substrate trapping as a function of the instantaneous Vds and Vdsq . The surface-trapping effect can be captured by setting the dynamic I −V characteristics at the static point of Vdsq = 0 V and the varied static gate–source bias. It follows the assumption that the surface-trapping effect is mainly function of the gate–source bias voltage due to it will reduce the effectiveness of the applied Vgs . In a similar manner, the backgate voltage caused by substrate trapping effect [31] can be characterized by measuring the pulsed I−V at different bias points of drain–source voltages but with constant gate–source voltage. Fig. 8 shows the measured and simulated pulsed I −V behavior at different bias to capture the charge-trapping effect. The pulsed I−V measurement biased at Vgsq /Vdsq = 0/0 V

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Fig. 8. (a) Measured (circles) and simulated (lines) pulsed I−V characteristics at biases (a) Vgsq = 0 V and Vdsq = 0 V, (b) Vgsq = −4 V and Vdsq = 0 V, (c) Vgsq = −4 V and Vdsq = 35 V, and (d) Vgsq = −2.5 V and Vdsq = 28 V for Vgs = −4 to 0 V, 0.2-V steps and Vds = 0 to 35 V, 1-V step.

in Fig. 8(a), which has negligible trapping and self-heating effects, is considered as a reference. The comparison of Ids characteristics at different quiescent points with Vgsq = 0 V and Vdsq = 0 V [Fig. 8(a)] and Vgsq = −4 V and Vdsq = 0 V [Fig. 8(b)] can illustrate the reduction of effectiveness of Vgs caused by the surface traps. It is manifest that the value of Ids in Fig. 8(b) is smaller than Fig. 8(a). On the other hand, the influence of back-gate voltage as a result of substrate trapping is illustrated by the difference between PIV behavior at Vgsq = −4 V and Vdsq = 0 V [Fig. 8(b)] and Vgsq = −4 V and Vdsq = 35 V [Fig. 8(c)]. The collective influence of surface and substrate effects is examined by PIV measurement bias at Vgsq = −2.5 V and Vdsq = 28 V [Fig. 8(d)]. From the close agreements between modeled and measured results, it is clear that the proposed dispersive model established through introducing into effective gate–source voltage can accurately predict the pulsed I−V characteristics. III. M ODEL V ERIFICATION The developed large-signal model is embedded by using the SDD in spice-like software Keysight advance design systems (v.2013). The devices are measured by using the on-wafer load–pull system (Focus/MPT-3620-TC) for validation purpose. The proposed scalable model is validated by means of several 0.25 − μm AlGaN/GaN HEMTs with different geometries 4 × 50 μm, 4 × 100 μm (reference), and 6×100 μm. Fig. 9 shows the photographs of 4×100 μm reference device and test block diagram for device characterization used for model verification. A. Small-Signal Characterization Before verifying the large-signal characteristics, the smallsignal S-parameter should be validated first to ensure the consistency between small- and large-signal models. In Fig.10,

Fig. 9. (a) Photograph of AlGaN/GaN HEMT device with 4 × 100 μm. (b) Test block diagram for device characterization used for model verification.

the measured and modeled scattering parameters of three verified devices are shown over the 0.1- to 40-GHz frequency range at two different bias Vgs = −1.5 V and Vds = 10 V and Vgs = −2.5 V and Vds = 28 V. The good agreements between the simulations and measurements demonstrate that the proposed scalable SP model with the nonlinear geometry and dissipated power dependent thermal resistance and appropriate scaling rules can accurately predict the small-signal performance for different device geometries. However, there exist slight deviations in the low-frequency S22 due to the bias-dependence of Cds is not taken account into the model. Moreover, the kink effect is observed in S22 for these three GaN HEMTs and it gets more pronounced for larger devices and higher bias due to the higher transconductance [32]. B. Multiharmonic Load–Pull Results In order to verify the accuracy of the model predicting the maximum output power or high efficiency under different harmonic terminations, the load–pull measurements and simulations under different harmonic impedances have been performed. The values of the optimal impedances at Z f 0 , Z 2 f 0 , and Z 3 f 0 of the 4 × 50, 4 × 100, and 6 × 100 μm AlGaN/GaN HEMTs have been shown in Table III. A verification of the harmonic output power predicted by the proposed scalable model is demonstrated in Fig. 11. A good agreement can be obtained for the second- and third-harmonic power which means that the proposed model can accurately predicted multiharmonic performance under different load impedances at 8 GHz for the three devices. This can be attributed to the accurate drain current model which could be largely related to the generation of harmonics.

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Fig. 11. Measured (symbols) and simulated (lines) harmonics Pout at f 0 = 8 GHz, Vgs = −2.5 V, and Vds = 28 V. (a) 4 × 50 μm. (b) 4 × 100 μm. (c) 6 × 100 μm.

Fig. 10. Measured (symbols) and modeled (solid lines) S-parameters for 0.1 to 40 GHz at Vgs = −1.5 V and Vds = 10 V (left) and Vgs = −2.5 V and Vds = 28 V (right). (a) 4 × 50 μm. (b) 4 × 100 μm. (c) 6 × 100 μm. TABLE III O PTIMAL L OAD I MPEDANCES

C. Large-Signal Characterizations To obtain the optimal output performance, the manipulations of the fundamental and high-order harmonics load terminations are carried out, which is significant for the design of power amplifier to acquire maximum output power or the highest efficiency. Through the realized load terminations, the optimum large-signal output performance has been achieved. The comparison between measurements and simulations of the fundamental output power (Pout ), gain and PAE under the conditions terminated for maximum power-added efficiency is shown in Fig. 12. The devices with 4 × 50, 4 × 100, and 6 × 100 μm gate width are measured and simulated at 8 GHz which are shown in Fig. 12(a)–(c), respectively. The good

Fig. 12. Single-tone power sweep simulations (lines) and measurements (symbols) for power characteristics (Pout , gain, and PAE versus input power Pin ) of three different gate geometries AlGaN/GaN HEMTs for Vgs = −2.5 V and Vds = 28 V at 8 GHz. (a) 4 × 50 μm. (b) 4 × 100 μm. (c) 6 × 100 μm.

agreement between measurements and simulation indicates that the proposed model can not only accurately predict the large-signal behavior under the optimum load impedances but also be used for different device geometries with good accuracy. IV. C ONCLUSION In this paper, an improved scalable SP large-signal model of AlGaN/GaN HEMTs suitable for multiharmonic characteristics with different device periphery is presented. The selfheating and charge-trapping effects are incorporated into the

WU et al.: SCALABLE MULTIHARMONIC SP MODEL

Ids model with good accuracy. The scaling rules for differentin-size devices are also presented. The scalable thermal resistance is accurately identified by means of electrothermal FEM simulations which is easily scalable with the gate width and power dissipations. The accuracy and effectiveness of model are validated by dc-I−V , pulsed I−V , and small- and largesignal performance by using 0.25 − μm AlGaN/GaN HEMTs with different peripheries. The results show that accurate predictions, especially under the conditions of different load impedances by using the proposed scalable large-signal model are achieved. This scalable model can not only be suitable for device characterizations, but also be applied to accurately optimize device structure for the desired performance in MMIC power amplifiers design. R EFERENCES [1] U. K. Mishra, L. Shen, T. E. Kazior, and Y.-F. Wu, “GaN-based RF power devices and amplifiers,” Proc. IEEE, vol. 96, no. 2, pp. 287–305, Feb. 2008. [2] R. P. Smith et al., “AlGaN/GaN-on-SiC HEMT technology status,” in Proc. IEEE Compound Semiconductor Integr. Circuits Symp., Oct. 2008, pp. 1–4. [3] R. S. Pengelly, S. M. Wood, J. W. Milligan, S. T. Sheppard, and W. L. Pribble, “A review of GaN on SiC high electron-mobility power transistors and MMICs,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1764–1783, Jun. 2012. [4] Y. J. Qiu, Y. H. Xu, R. M. Xu, and W. G. Lin, “Compact hybrid broadband GaN HEMT power amplifier based on feedback technique,” Electron. Lett., vol. 49, no. 5, pp. 372–374, 2013. [5] W. L. Fu, Y. H. Xu, B. Yan, B. Zhang, and R. M. Xu, “Numerical simulation of local doped barrier layer AlGaN/GaN HEMTs,” Superlattices Microstruct., vol. 60, pp. 443–452, Aug. 2013. [6] D. Mari, M. Bernardoni, G. Sozzi, R. Menozzi, G. A. Umana-Membreno, and B. Nener, “A physical large-signal model for GaN HEMTs including self-heating and trap-related dispersion,” Microelectron. Rel., vol. 51, no. 2, pp. 229–234, 2011. [7] T. M. Roh, Y. Kim, Y. Suh, W. S. Park, and B. Kim, “A simple and accurate MESFET channel-current model including bias-dependent dispersion and thermal phenomena,” IEEE Trans. Microw. Theory Techn., vol. 45, no. 8, pp. 1252–1255, Aug. 1997. [8] J. Wang, L. Sun, J. Liu, and M. Zhou, “A surface-potential based compact model of gate capacitance in GaN HEMTs,” in Proc. Prog. Electromagn. Res. Symp., 2013, pp. 250–254. [9] S. Khandelwal and T. A. Fjeldly, “A physics based compact model of I −V and C−V characteristics in AlGaN/GaN HEMT devices,” SolidState Electron., vol. 76, pp. 60–66, Oct. 2012. [10] X. Cheng and Y. Wang, “A surface-potential-based compact model for AlGaN/GaN MODFETs,” IEEE Trans. Electron Devices, vol. 58, no. 2, pp. 448–454, Feb. 2011. [11] W. Deng, J. Huang, X. Ma, and J. J. Liou, “An explicit surface potential calculation and compact current model for AlGaN/GaN HEMTs,” IEEE Electron Device Lett., vol. 36, no. 2, pp. 108–110, Feb. 2015. [12] M. Patrick and L. Luca, “A compact model of AlGaN/GaN HEMTs power transistors based on a surface-potential approach,” in Proc. Int. Conf. Mixed Design Integr. Circuits Syst., Jun. 2013, pp. 92–95. [13] Q. Wu, Y. Xu, Z. Wen, Y. Wang, and R. Xu, “A surface potential large signal model for AlGaN/GaN HEMTs,” in Proc. 11th EuMIC, London, U.K., Oct. 2016, pp. 349–352. [14] Q. Wu, Y. Xu, Z. Wang, L. Xia, B. Yan, and R. Xu, “Implementation of self-heating and trapping effects in surface potential model of AlGaN/GaN HEMTs,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, USA, Jun. 2017, pp. 236–239. [15] C. Wang et al., “An electrothermal model for empirical large-signal modeling of AlGaN/GaN HEMTs including self-heating and ambient temperature effects,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 2878–2887, Dec. 2014. [16] Z. Wen, Y. Xu, C. Wang, X. Zhao, and R. Xu, “An efficient parameter extraction method for GaN HEMT small-signal equivalent circuit model,” Int. J. Numer. Model., Electron. Netw., Devices Fields, vol. 30, no. 1, p. e2127, Jan./Feb. 2015, doi: 10.1002/jnm.2127.

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[17] G. Dambrine, A. Cappy, F. Heliodore, and E. Playez, “A new method for determining the FET small-signal equivalent circuit,” IEEE Trans. Microw. Theory Techn., vol. MTT-36, no. 7, pp. 1151–1159, Jul. 1988. [18] P. M. White and R. M. Healy, “An improved equivalent circuit for determination of MESFET and HEMT parasitic capacitances from ‘Cold-FET’ measurements,” IEEE Microw. Guided Wave Lett., vol. 3, no. 12, pp. 453–454, Dec. 1993. [19] A. Jarndal and G. Kompa, “A new small-signal modeling approach applied to GaN devices,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 11, pp. 3440–3448, Nov. 2005. [20] D. E. Ward, “Charge-based modeling of capacitance in MOS transistors,” Integr. Circuits Lab., Stanford Univ., Stanford, CA, USA, Tech. Rep. G201-11, Jun. 1981. [21] A. Jarndal and G. Kompa, “An accurate small-signal model for AlGaN-GaNHEMT suitable for scalable large-signal model construction,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 333–335, Jun. 2006. [22] J.-W. Lee, S. Lee, and K. J. Webb, “Scalable large-signal device model for high power-density AlGaN/GaN HEMTs on SiC,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 679–682. [23] Y. Xu et al., “A scalable GaN HEMT large-signal model for highefficiency RF power amplifier design,” J. Electromagn. Waves Appl., vol. 28, no. 15, pp. 1888–1895, 2014. [24] D. Resca, A. Raffo, A. Santarelli, G. Vannini, and F. Filicori, “Scalable equivalent circuit FET model for MMIC design identified through FW-EM analyses,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 2, pp. 245–253, Feb. 2009. [25] A. Nalli et al., “GaN HEMT noise model based on electromagnetic simulations,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 8, pp. 2498–2508, Aug. 2015. [26] G. Sozzi and R. Menozzi, “A review of the use of electro-thermal simulations for the analysis of heterostructure FETs,” Microelectron. Rel., vol. 47, no. 1, pp. 65–73, 2007. [27] H. Hjelmgren, M. Thorsell, K. Andersson, and N. Rorsman, “Extraction of an electrothermal mobility model for AlGaN/GaN heterostructures,” IEEE Trans. Electron Devices, vol. 59, no. 12, pp. 3344–3349, Dec. 2012. [28] K. Shirakawa, M. Shimizu, Y. Kawasaki, Y. Ohashi, and N. Okubo, “A new empirical large-signal HEMT model,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 4, pp. 622–624, Apr. 1996. [29] O. Jardel et al., “An electrothermal model for AlGaN/GaN power HEMTs including trapping effects to improve large-signal simulation results on high VSWR,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 12, pp. 2660–2669, Dec. 2007. [30] M. Chaibi et al., “Nonlinear modeling of trapping and thermal effects on GaAs and GaN MESFET/HEMT devices,” Prog. Electromagn. Res., vol. 124, no. 1, pp. 163–186, 2012. [31] K. S. Yuk and G. R. Branner, “An empirical large-signal model for SiC MESFETs with self-heating thermal model,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 11, pp. 2671–2680, Nov. 2008. [32] G. Crupi, A. Raffo, A. Caddemi, and G. Vannini, “Kink effect in S22 for GaN and GaAs HEMTs,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 5, pp. 301–303, May 2015.

Qingzhi Wu was born in Hei Longjiang, China. She received the B.S. degree from the University of Electronic Science and Technology of China, Chengdu, China, in 2013, where she is currently pursuing the Ph.D. degree in electromagnetic field and microwave techniques. Her current research interests include microwave third-generation semiconductor power devices modeling, with particular focus on the physics-based modeling method and applications for GaN HEMTs.

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Yuehang Xu (M’11–SM’16) received the B.S. and M.S. degrees in electromagnetic field and microwave techniques from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2004 and 2007, respectively, and the Ph.D. degree from UESTC joint with Columbia University, New York, NY, USA, in 2010. In 2016, he joined Case Western Reserve University, Cleveland, OH, USA, as a Visiting Associate Professor. He joined the Department of Electronic Engineering, UESTC, in 2010, where he has been a Professor since 2017. He has authored or co-authored more than 150 scientific papers in international journals and conference proceedings. His current research interests include modeling and characterization of radio frequency micro-/nanoscale electronic devices and MMIC design. Yongbo Chen was born in Sichuan, China, in 1985. He received the B.S. and Ph.D. degrees in electromagnetic field and microwave techniques from the University of Electronic Science and Technology of China, (UESTC) Chengdu, China, in 2007 and 2013, respectively. From 2011 to 2013, he was a Visiting Scholar with the Beckman Institute of University of Illinois, Urbana–Champaign, Urbana, IL, USA. He is currently a Post-Doctoral Researcher with UESTC. His current research interests include modeling of semiconductor devices and monolithic microwave integrated circuit designs. Yan Wang received the B.S. and M.S. degrees in electrical engineering from Xian Jiaotong University, Xian, China, in 1988 and 1991, respectively, and the Ph.D. degree in semiconductor device and physics from the Institute of Semiconductors, Chinese Academy of Science, Beijing, China, in 1995. Since 1999, she has been a Professor with the Institute of Microelectronics, Tsinghua University, Beijing. Her current research interests include device modeling.

Wenli Fu was born in Shanxi, China, in 1988. She received the B.S. and M.S. degrees in electromagnetic field and microwave techniques from the University of Electronic Science and Technology of China, Chengdu, China, in 2010 and 2013, respectively. Her current research interests include microwave power transmission technology and MMIC design.

Bo Yan received the B.S. and M.S. degrees in electromagnetic field and microwave techniques, and the Ph.D. degree from the University of Electronic Science and Technology of China, Chengdu, China, in 1991, 1998, and 1998, respectively. His current research interests include microwave and millimeter-wave hybrid integrated circuit and MCM technology.

Ruimin Xu (M’07) was born in Sichuan, China, in 1958. He received the B.S. and Ph.D. degrees in electromagnetic field and microwave techniques from University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1982 and 2007, respectively. He is currently a full Professor with UESTC. His current research interests include microwave and millimeter-wave technologies and applications, and radar systems.

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A Passive PEEC-Based Micromodeling Circuit for High-Speed Interconnection Problems Yuhang Dou, Student Member, IEEE, and Ke-Li Wu , Fellow, IEEE

Abstract— A passive partial element equivalent circuit (PEEC)-based micromodeling circuit is proposed for time-domain simulation of a high-speed interconnection problem. This physics-based model order reduction method derives a concise and physically meaningful circuit model from the PEEC model by absorbing its insignificant nodes. To maintain high fidelity of the original electromagnetic PEEC model, the concept of pseudoinductor is introduced to the nodeabsorbing process. The derivation process does not involve any matrix inversion or decomposition and is highly suitable for GPU parallel computations. Passivity of the micromodeling circuit is ensured by a new passivity checking and enforcement method proposed for the first time. As the scale of the micromodeling circuit can be one order of magnitude smaller than that of the original PEEC model, the time-domain simulation can be three orders of magnitude faster. Two practical examples are given to demonstrate the high fidelity, scalability, and accuracy of the proposed micromodeling circuit, showing excellent applicability to high-speed interconnection problems. Index Terms— Electromagnetic (EM) modeling, equivalent circuit, model order reduction (MOR), partial element equivalent circuit (PEEC), signal integrity (SI).

I. I NTRODUCTION

S

IGNAL integrity (SI) is a set of measures of the quality of an electrical signal. A digital signal with good SI must have stable and valid logic levels; accurate placement in time; clean and fast transitions; and be free of any transients [1]. Nowadays, it is increasingly critical to ensure a good SI design of interconnection and packaging circuits for highspeed digital signals [2] with the continuous increase of data rate and circuit density. Usually, eye diagram is used as a visual indicator to observe the general SI on a clocked bus, for which passivity of the model simulated in time-domain is required and simulation time must be bearable. The partial element equivalent circuit (PEEC) model [3] converts a multiconductor electromagnetic (EM) problem into a circuit model, which can be solved by Modified Nodal Analysis (MNA) [4] or a SPICE-like solver. It has been widely adopted in SI analysis, electronic packaging design,

Manuscript received August 15, 2017; revised October 28, 2017; accepted November 15, 2017. This work was supported by the University Grants Committee of Hong Kong under Grant AoE/P-04/08. (Corresponding author: Ke-Li Wu.) The authors are with the Department of Electronic Engineering, Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2779484

EM radiation, EM compatibility, and power electronics problems [5]–[9]. For a large and complex interconnection problem, the PEEC model consists of an excessively large number of partial elements, whose simulation time is prohibitively long because multiple large-scale matrix inversions are involved. Besides, it is difficult for the designers to acquire any physical insight by examining the massive discretization meshbased partial elements. Nonetheless, a PEEC model can serve as a good starting point to derive a much concise and physically meaningful circuit for frequency-domain simulations [10]–[12]. However, these micromodeling circuits cannot be used for time-domain simulation due to their poor passivity conditions. To facilitate time-domain analysis of system responses of a large-scale EM problem, various model order reduction (MOR) methods have been developed since the 1990s. The methods can be classified into two categories according to their objectives: macromodeling and micromodeling. The macromodeling attempts to extract a concise model represented by a set of state-space equations while preserving the input–output characteristics of the original problem [13]. Usually, a macromodel does not have a direct correspondence with the physical layout of the problem. On the other hand, the micromodeling provides a circuit domain representation that comes with certain physical meaning. The proposed PEEC-based micromodeling circuit finds an RLC circuit that describes the EM effects of a physical interconnection problem [10]–[12]. Many macromodeling methods [13]–[20] have been developed. The well-known Lanczos and Arnoldi algorithms, a subclass of the so-called Krylov subspace methods, constitute the framework of modern MOR research. The main idea of these mathematics-based MOR methods is to find a projection of a large-scale state-space system onto a lower dimensional subspace by a set of appropriate basis vectors [20]. Although the projected lower order state-space system can reserve certain mathematic properties, such as passivity, causality, and reciprocity, it lacks direct physical interpretation of the original EM problem. The Micromodeling methods that are based on node elimination include AMOR [21], TICER [22], and SIP [23]. The AMOR method [21] is based on observation that those adjacent nodes of the RC circuits with almost the same voltage can be aggregated together as a “super node,” which is not suitable for PEEC circuit with coupled RLC elements. The TICER approach [22] uses the Y− transformation to reduce

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a general RLC circuit including inductive couplings for the first time. Similar concept is independently developed for a general PEEC circuit with coupled inductors and coupled capacitors for embedded RF passive designs [10], [11]. The SIP method [23] for RLC circuit is a projection method mathematically, but its implementation is similar to the node eliminationbased method. However, these methods will introduce a large amount of redundant branches during node eliminate process, and require two large-scale matrix inversions in order to obtain a reducible inductance and capacitance matrices from the PEEC model. Recently, a direct mesh-based micromodeling method for the PEEC model was proposed [12]. The method overcomes two critical issues faced with the Y- transformation-based micromodeling methods: 1) the scalability issue constrained by matrix inversions of large-order inductance and potance matrices and 2) the proxy direct inductance for the actual mutual inductances of the original problem. In the micromodeling circuit [12], the circuit nodes are in correspondence with the physical layout and the accuracy in terms of frequencydomain responses can be specified by the user. However, the micromodeling circuit in [12] may generate negative selfinductances due to missing some critical inductive components in the circuit transformation, which causes instability issues in time-domain simulation. In this paper, an improved micromodeling method is proposed to overcome the nonpassivity problem. Different from the previous micromodeling methods, the proposed method introduces a pseudoinductor for each grounded potors to retain the inductive components more accurately in the MOR process. At the end of the micromodeling process, the pseudoinductors will be internalized by regular inductors. It is demonstrated through numerical examples that by introducing the pseudoinductor, the improved micromodeling circuit is in higher conformity with the original PEEC model than the existing micromodeling circuits. The numerical results show that the proposed micromodeling circuit ensures all the selfinductances to be positive, and the passivity violation factor of the proposed micromodeling circuit is smaller than that of the existing method [12] by three orders of magnitude. Most importantly, a new passivity check and enforcement method for a general RLC circuit is proposed to systematically correct the numerical error in a nonpassive circuit model so that the passivity of the circuit model can be guaranteed for timedomain simulation. In addition to maintain higher fidelity of the physical meaning of an EM problem in the circuit model, this improved micromodeling method inherits the two most attractive attributes from [12]: 1) high scalability, i.e., there is no matrix inversion or decomposition involved in the modeling process and 2) high suitability of being accelerated by the GPU parallel computation. In this paper, the rigorous derivation of the proposed micromodeling method, the passivity check and enforcement method and the scheme for the GPU parallel computation will be discussed in detail. The numerical results demonstrate that the computational time for SI analysis of a practical high-speed interconnection problem using the proposed micromodeling

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Fig. 1.

PEEC meshes and their circuit representation.

circuit is about three orders of magnitude less than that if a traditional PEEC model is used. Without loss of generality, this paper only concerns the quasi-static EM problems, in which the electrical size of the circuit cannot be too large so that the radiation effect can be neglected. II. PEEC R EPRESENTATION OF AN EM P ROBLEM The PEEC model is a circuit representation of the discretized MPIE [3]. The quasi-static MPIE for infinitely thin conductors can be expressed as      E(r) = − j ω G A (r, r ) · J(r )ds − ∇ G ϕ (r, r )ρ(r )ds  s

s

(1) where G¯¯ A and G ϕ are Green’s functions of magnetic vector and electric scalar potentials, respectively; and ρ and J are the surface charges and current densities, respectively. Without loss of generality, only the x-component in (1) is considered. Discretizing the conductor surface for the current and charge densities using PEEC meshes, as shown in Fig. 1, then applying Galerkin’s matching procedure on an inductive mesh l, the discretized form of (1) is given by      ll x  j ω G xAx r, r dsm dsl Imx Il + σ wl wl wm  m 

    G ϕ r, r dsn dsl − G ϕ r, r dsn dsl + Qn = 0 + − al + an al − an n (2) where mesh m, l are inductive meshes and w is the width of an inductive mesh; mesh n, l − and l + are capacitive meshes and a is the area of a capacitive mesh; and indexes l − and l + are two terminals of inductive cell l with current Il flowing from node l − to node l + . A finite-difference approximation has been used for the derivative operator appearing in the third term of (2). In a circuit-oriented form, (2) is represented as   Rl Il + j ωMl,m Im + (Pl + ,n − Pl − ,n )Q n = 0 (3) m

n

where constant Ml,m is the partial (self- or mutual) inductance between inductive cells l and m; Im is the current flowing through the inductive cell m; constant Pl ± n is the coefficient of electric potential between capacitive cells l ± and n; and Q n is the total charge on the capacitive cell n. For ease of the micromodeling, conductor loss Rl is conflated with the

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corresponding self-inductance Ml,m and Q n is replaced by In /j ω, (3) becomes   j ωMl,m Im + (Pl + ,n − Pl − ,n )In /j ω = 0. (4) m

n

In an analogous manner to conventional definitions of selfinductance of an inductor and mutual inductance of two inductors, quantity Pi,i is called as self-potance of the circuit element potor i and Pi, j is called as mutual potance of two potors i and j . Equation (4) can be interpreted by the circuit in Fig. 1, in which inductors are coupled with each other and potors are coupled with each other as well. For clarity, mutual inductance and mutual potance are represented by circular dots and square dots, respectively. The proposed micromodeling method deals with potance matrix and inductance matrix directly. III. T HEORY O F M ICROMODELING M ETHOD A good micromodeling method based on the PEEC model should possess the following three desired attributes: 1) it must be passive; 2) it contains as much physical essence of the original EM problem as possible; and 3) it is highly suitable for GPU parallel computation. However, the micromodeling method in [12] proposed for frequency-domain analysis is not favoring in attribute 1 due to its inevitable negative self-inductances. This issue is overcome by this improved micromodeling method. Fig. 2 outlines the micromodeling process. Fig. 2(a) shows a conventional PEEC model. In step 1, pseudoinductors are introduced to the PEEC model, as shown in Fig. 2(b), which are critical to preserve accurate inductive components in the micromodeling process. In step 2, a recursive process for absorbing an insignificant node is conducted. The process first finds the most insignificant node (MIN), say, node k in Fig. 2(b), in a low-pass sense, then the node is absorbed by a new equivalent circuit transformation, as illustrated in Fig. 2(c). Having had the node absorbed, the shunt branches generated in the circuit transformation need to be combined, as illustrated in Fig. 2(d). Repeating step 2 until all insignificant nodes are absorbed and a concise circuit model is obtained as the one shown in Fig. 2(e). In step 3, the pseudoinductors are internalized by surrounding regular inductors, as shown in Fig. 2(f). Finally, in step 4, the passivity of the micromodeling circuit is checked and enforced if necessary. As a result, a passive and concise micromodeling circuit is obtained. The theories for the fourstep process are explicated in the following sections. A. Introducing Pseudoinductors for PEEC Model The proposed method adds a pseudoinductor in series with each grounded potor at the beginning of the micromodeling process, as shown in Fig. 2(b). To distinguish with the pseudoinductors, the inductors that are defined by inductive meshes are called regular inductor. Having had the pseudoinductors introduced in the process of absorbing the insignificant node k, as shown in Fig. 2(b), the voltages of the neighboring nodes of node k are able to remain the same in the circuit transformation, as shown in Fig. 2(c). As a result, the method

Fig. 2. Illustration of the proposed micromodeling method. (a) Mesh and circuit topology of PEEC model. (b) Finding the MIN k. (c) Circuit transformation to absorb node k. (d) Combining coupled shunt branches. (e) Mesh and circuit topology after absorbing all insignificant nodes. (f) Internalizing pseudoinductors by regular inductors.

preserves more physical essence in the node absorbing process than that in [12]. The initial values of pseudoinductors are set to zero and they will be updated after absorbing each insignificant node. In the end of micromodeling, these pseudoinductors

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4) All mutual couplings associated with the assimilated elements are transferred to the mutual couplings with the newly added elements. 5) Unchanged potors and pseudoinductors are named as other potors and other inductors, respectively. Their associated PpI branches are called other branches. 6) Unchanged regular inductors are named as remaining inductors. The notations and voltage–current relations of the circuit variables in the circuit transformation are defined as follows. 1) The potance matrix P( p) before node k being absorbed ( p) ( p) is defined by voltage vector v p and current vector i p as 





 ( p) ( p) ( p) ( p) 1 PO O pO A vP O iO =   ( p) ( p) j ω p(Op)A T p(Ap)A vPA iA       ( p)

P( p)

vP

Fig. 3. Circuit transformation for absorbing an insignificant node. (a) Circuit prior to transformation. (b) Circuit posterior to transformation.

( p)

iP

where subscripts O and A refer to the other and the assimulated potors, respectively. 2) The inductance matrix M( p) before node k being ( p) absorbed is defined by voltage vector v M and current ( p) vector i M as ⎡  ( p) T ( p) ⎤ ( p) MR R MO R vM R ⎢ ( p) ⎥ ⎢ ( p) ( p) MO O ⎢v M O ⎥ ⎢ MO R ⎢ ( p) ⎥ = j ω ⎢ ( p) ⎣ v MC ⎦ ⎣ MC( p)R MC O  ( p) T  ( p) T ( p) vM A mR A mO A    ⎡

will be internalized by regular inductors as shown in Fig. 2(f), because the pseudoinductors are actually contributed by the regular inductors. For convenience, the branch of a potor serially connected with a pseudoinductor is called as a potor-pseudoinductor branch, or a PpI branch.

( p)

vM

 ( p) T M  (Cp)R T MC O ( p) M  ( CC p) T mC A

( p) ⎤ mR A ( p) ⎥ mO A ⎥ ( p) ⎥ mC A ⎦

( p) ⎤ iR ⎢ ( p) ⎥ ⎢i ⎥ × ⎢ (Op) ⎥ ⎣ iC ⎦

( p)

m AA

M( p)

⎡

B. Hypotheses of Circuit Transformation The new circuit transformation concerns absorbing a node in accordance with a physical sensible remeshing scheme, as illustrated in Fig. 2(b) and (c). Assuming that node k in Fig. 3(a) is the MIN to be absorbed at the pth step of the recursive node absorbing process, following hypotheses and definitions are given, referring to Fig. 3(a) and (b) with node k being absorbed. 1) Assume the potor and pseudoinductor associated with node k as well as all the inductors connecting to node k are to be assimilated; they are called assimilated potor, assimilated inductor, and connecting inductors, respectively. 2) An incremental PpI branch is added to each of the Nc neighboring nodes to share one-Ncth portion of the potance of mesh k. The potors and inductors in the added PpI branches are called incremental potors and incremental inductors, respectively. 3) A new inductor is introduced between each consecutive pair of the neighboring nodes, reflecting the newly introduced current paths. The newly introduced inductors are called new inductors.

(5)



(6)

( p)

iA   ( p)

iM

where subscripts R, O, C, and A refer to the remaining inductors, the other inductors, the connecting inductors, and the assimilated inductors, respectively. 3) The potance matrix P ( p) after node k being absorbed is defined by voltage vector v (pp) and current vector i (pp) as 

   ( p) T   ( p)

( p) 1 P (Op)O v P O iO P IO = ( p) ( p)  ( p) j ω P (I p) v P I i I P O II       ( p)

v P

P ( p)

(7)

( p)

i P

where subscripts O and I refer to the other potors and the incremental potors, respectively. 4) The inductance matrix M ( p) after node k being absorbed ( p) ( p) is defined by voltage vector v M and current vector i M

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as ⎡

( p) v M R ⎢  ( p) ⎢ v MO ⎢  ( p) ⎣ v MN ( p) v M I





( p)

5

⎤

D. Circuit Transformation

⎥ ⎥ ⎥ ⎦

The following discussions derives the proposed circuit transformation. 1) Updating Formulas Related to Incremental Elements: According to (7) and (8), the voltages across the incremental PpI branches are



v M

⎡

( p)

M R R ⎢  ( p) ⎢M = j ω⎢  (Op)R ⎣M N R ( p) M I R 



( p) T   ( p) T   ( p) T ⎤ ⎡  ( p) ⎤ M O R M M R i   (Np)R T   (I p) T ⎥ ⎢ (Rp) ⎥ ( p) M NO M I O ⎥ ⎢i O ⎥ M O O   ( p) T ⎥ ⎢  ( p) ⎥ ( p) ( p) M N O M N N M I N ⎦ ⎣i N ⎦ ( p) ( p) ( p) ( p) i I M I O M I N M I I    ( p) i M

M ( p)

(8) where subscripts R, O, N, and I refer to the remaining inductors, the other inductors, the new inductors and the incremental inductors, respectively. 5) Let a be a column vector, whose i th element is a(i ). Define the N-by-1 vector a¯ that is related to the N-by1 vector a by a(i ¯ ) = a(i %N + 1) (9) where modulo operation i %N finds the remainder after division of i by N. For example, 1%2 = 1 and 2%2 = 0. 6) Let B be a matrix, whose element in i th row and j th column is expressed as b(i, j ). Define N-by-N matrices ¯ B˜ and Bˆ that are related to an N-by-N matrix B by B, ¯ j ) = b(i %N + 1, j ) b(i, ˜ j ) = b(i, j %N + 1) b(i, ˆ j ) = b(i %N + 1, j %N + 1). b(i, (10) 7) Let j N be an N-by-1 vector and J N be an N-by-N matrix, whose elements are all one. Let I N be an Nby-N identity For  matrix.   example    1 1 1 1 0 J2 = I2 = . j2 = 1 1 1 0 1 (11) C. Equivalence for the Circuit Transformation The following equivalent conditions are imposed on the proposed circuit transforamtion: 1) Referring Fig. 3(a) and (b), equivalent KVL and KCL conditions for the transformation are ( p) v P I

¯i( p) − N

( p) + v M I ( p) v M N ( p) ( p) i N + i¯ I ( p) jTNC i I ( p) NC i I

= = = = =

( p) ( p) v MC + j NC v M A ( p) ( p) v MC − v¯ MC ¯i( p) C ( p) iA ( p) i A j NC .

( p) + j NC v P A

(12a) (12b) (13a) (13b)

( p)

( p)

( p) i O

=

( p) iO

( p)

( p)

v M O = v M O

( p) i R

=

( p) iR .

( p)

(15a) According to (12a), (5), and (6), one can further find that ( p)

( p)

v M I + v P I ( p)

( p)

( p)

= v MC + j NC v M A + j NC v P A  ( p) ( p) ( p) ( p) ( p) ( p) ( p) ( p)  = j ω MC R i R + MC O i O + MCC iC + mC A i A  ( p) T ( p)  ( p) T ( p)  ( p) T ( p) + j ωj NC m R A i R + m O A i O + mC A iC    ( p) T ( p) ( p) ( p)  ( p) ( p)  +m A A i A + 1/j ω j NC p O A i O + p A A i A . (15b) By using (13), (15b) can be expressed as ( p)

( p)

v M I + v P I  ( p) T  ( p)  ( p) = j ω MC R + j NC m R A i R  ( p)  ( p) T  ( p) + j ω MC O + j NC m O A i O    ( p) T   ( p)  ( p) ˜ ( p) + j NC m( p) T − j NC m ¯ CA i N + j ω MCC − M CC CA  ( p) T  ( p)  ( p) ( p) T ( p) + j ω MCC +j NC mC A +mC A j NC + NC m A A I NC i I    ( p) T ( p) ( p) ( p)  + 1/j ω j NC p O A i O + NC p A A I NC i I . (15c) Comparing the right-hand sides of (15a) and (15c), the updating formulas for the circuit elements related to the incremental elements are obtained as  ( p) T ( p) ( p) M I R = MC R + j NC m R A  ( p) T ( p) ( p) M I O = MC O + j NC m O A     ( p) ( p) ˜ ( p) + j NC m( p) T − j NC m¯( p) T M I N = MCC − M CC CA CA  ( p) T ( p) ( p) ( p) ( p) M I I = MCC +j NC mC A +mC A jTNC + NC m A A I NC  ( p) T ( p) P I O = j NC p O A ( p) P I I

=

( p) NC p A A I NC .

(16a) (16b) (16c) (16d) (16e) (16f)

(13c)

2) The voltages and currents across and through the other potors, other inductors and remaining inductors are unchanged, that is, v P O = v P O

( p)

v M I + v P I  ( p) ( p) ( p) ( p) ( p) ( p) ( p) ( p)  = j ω M I R i R + M I O i O + M I N i N +M I I i I   ( p) ( p) ( p) ( p)  + 1/j ω P I O i O + P I I i I .

( p)

( p)

v M R = v M R (14a) (14b)

2) Updating Formulas Related to New Inductors: According to (8), the voltages of the new inductors are ( p)

v M N

 ( p) ( p) ( p) ( p) ( p) ( p)  ( p) T ( p)  = j ω M N R i R +M N O i O +M N N i N + M I N i I . (17a)

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According to (12b), the above voltages can be expressed as ( p)

( p)

( p)

v M N = v MC − v¯ MC  ( p) ( p) ( p) ( p) ( p) ( p) ( p) ( p)  = j ω MC R i R + MC O i O + MCC iC + mC A i A  ( p) ( p) ( p) ( p)  ¯ ( p) i( p) + M ¯ ( p) i( p) + m ¯ i +M . ¯ i − jω M CR R

CO O

CC C

CA A

(17b) Substituting (13a) and (13b), (17b) becomes  ( p)    ( p) ¯ ( p) i ( p) + j ω M( p) − M ¯ ( p) i ( p) v M N = j ω MC R − M CR R CO CO O   ( p) ˜ ( p) − M ¯ ( p) + M ˆ ( p) i ( p) + j ω MCC − M CC CC CC N  ( p)  ( p) ( p) ¯ ( p) + m( p) jTN − m + j ω MCC − M ¯ C A jTNC i I . CC CA C (17c) Comparing the right-hand sides of (17a) and (17c), the updating formulas of elements related to the new inductors are obtained as ( p) ( p) ¯ ( p) M N R = MC R − M CR ( p) ( p) ¯ ( p) M =M −M NO ( p) M N N ( p) M I N

= =

CO ( p) MCC ( p) MCC

(18a)

CO ˜ ( p) − M ¯ ( p) + M ˆ ( p) −M CC CC CC    ( p) T ˜ ( p) + j NC m( p) T − j NC m ¯ CA . −M CC CA

(18b) (18c) (18d)

( p)

Formulas (16c) and (18d) for updating M I N are the same, which shows that the proposed circuit transformation is symmetric. 3) Circuit Elements Related to Other Potors, Other Inductors, and Remaining Inductors: Using the same derivation method, it can be concluded that the other potors, other inductors, remaining inductors, as well as the mutual couplings between these circuit elements are not changed after the circuit transformation. That is, ( p)

( p)

( p)

( p)

( p)

( p)

( p)

( p)

P  O O = P O O M O O = M O O M R R = M R R M O R = M O R . (19) E. Combining Coupled Shunt Branches After the circuit transformation, each incremental PpI branch is in shunt with an other PpI branch and a new inductor branch is possibly in shunt with a remaining inductor branch. Every two shunt connected branches need to be combined as shown in Fig. 2(d) before absorbing the next insignificant node. Referring to Fig. 4, consider shunt connected branch S and branch I , which can be a pair of PpI branches or a pair of inductor branches and will be combined as branch S. The elements in the two branches are not only coupled to each other but are also coupled to the rest of circuit elements. PpI branches and inductor branches in the rest of circuit are distinguished by subscripts O and R, respectively. When the shunt branches are combined, all the mutual impedances in the circuit need to be updated. According to [12], the combined circuit elements as well as the concerned mutual couplings can be directly written here for the sack of brevity. 1) The self-impedance of combined branch S is  (20a) z S S = 1 yt .

Fig. 4. Combining process of two coupled shunt branches. (a) Circuit with two coupled shunt branches. (b) Circuit posterior to combining the coupled shunt branches.

2) The mutual impedances between combined branch S and remaining inductor branches or other PpI branches are  z R S = [(y S S + y S I ) z R S + (y I I + y S I ) z R I ] yt (20b)   z O S = [(y S S + y S I ) z O S + (y I I + y S I ) z O I ] yt . (20c) 3) The mutual impedances among other PpI branches and remaining inductor branches are y S S y I I + y S2 I (z O S − z O I ) (z O S −z O I )T yt (20d) 2 yS S y I I + yS I Z O R = Z O R − (z O S−z O I ) (z R S−z R I )T yt (20e) 2 yS S y I I + yS I Z R R = Z R R− (z R S−z R I ) (z R S−z R I )T yt (20f)   −1  z z yS S yS I = SS SI and yt = y S S + y I I +2y S I . where yS I y I I zSI zI I A general updating formula for inductance matrix and potance matrix can be derived from (20). Specific expressions for different types of shunt branches will be different. 1) Two Shunt Connected PpI Branches: When the shunt branches are PpI branches, the concerned inductance matrix and potance matrix of the circuit before combining can be expressed as ⎤ ⎡ M R R MTO R m R S m R I ⎢ MO R MO O mO S mO I ⎥ ⎥ M=⎢ T ⎦ ⎣ mT R S mO S m S S m S I T T m mO I m S I m I I ⎡ RI ⎤ PO O pO S pO I P = ⎣ pTO S p S S p S I ⎦ (21) pTO I p S I p I I Z O O = Z O O−

Similarly, the inductance matrix and potance matrix after combining can be expressed as ⎤ ⎡   T M OR m R S M R R ⎥ ⎢ M m O S ⎦ M = ⎣ M O R   T   O OT m RS m OS m S S    P O O p O S P  =   T . (22) p OS pS S

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7

Substituting (21) and (22) into (20) leads to the following relations: j ωm S S + pS S /j ω = j ωk1(e5 −e1 e4 e6 −ω2 e7 )+e1 e4 /jω 

j ωm O S j ωM O O

j ωm R S = j ωk2m R S + j ωk3 m R I + p O S /j ω = k2 ( j ωm O S + p O S /j ω)

m S S = e1 (e5 − e1 e4 e6 ) p S S = e1 e4

m  O S = e2 m O S + e3 m O I m  R S = e2 m R S + e3 m R I p O S = e2 p O S + e3 p O I P O O = P O O − e1 p O pTO

+ k3 ( j ωm O I + p O I /j ω)  + P O O /j ω = j ωM O O + P O O /j ω

M O O = M O O − e1 m O pTO − e1 p O mTO

− j ωk1( j ωm O + p O /j ω) × ( j ωm O + p O /j ω)T

M O R = M O R − e1 p O mTR M R R = M R R .

j ωM O R = j ωM O R − j ωk1( j ωm O + p O /j ω) × ( j ωm R )T j ωM R R = j ωM R R − j ωk1( j ωm R )( j ωm R )T (23) where c1 = ω2 (m S S + m I I − 2m S I )/( p S S + p I I − 2 p S I ); c2 = ω2 (m I I − m S I )/( p I I − p S I ); c3 = ω2 (m S S − m S I )/( p S S − p S I ); e1 = 1/( p S S + p I I − 2 p S I ), e2 = e1 ( p I I − p S I ); e3 = e1 ( p S S − p S I ), e4 = p S S p I I − ( p S I )2 ; e5 = p I I m S S + p S S m I I −2 p S I m S I , e6 = m S S +m I I −2m S I ; e7 = m S S m I I − (m S I )2 , k1 = e1 /(1 − c1 ); k2 = e2 (1 − c2 )/(1 − c1 ), k3 = e3 (1 − c3 )/(1 − c1 ); mO = mO S − mO I , mR = mR S − mR I , pO = pO S − pO I . By comparing the coefficients of ω and 1/ω terms of (23), the following updating formulas for the inductance and potance matrices that transform the circuit in Fig. 4(a) into (b) can be obtained: m S S = k1 (e5 − e1 e4 e6 − ω2 e7 ) p S S = e1 e4

m O S = k 2 m O S + k 3 m O I m R S = k 2 m R S + k 3 m R I p O S = k2 p O S + k3 p O I P O O = P O O − k1 p O pTO

M O O = M O O − k1 (m O pTO + p O mTO − ω2 m O mTO ) M O R = M O R − k1 (p O mTR − ω2 m O mTR ) M R R = M R R + ω2 k1 m R mTR .

and c3 , which are not in conformity with the original PEEC circuit. For a quasi-static problem, the three terms can be approximated to be zero in the low-pass sense. Therefore, the updated circuit elements become constants as

(24)

2) Two Shunt Connected Inductor Branches: The case of two shunt connected inductor branches is a special case of two shunt connected PpI branches when potances are absent. Therefore, updating formulas for inductance and potance matrices can be obtained from (24) as m S S = j ωg1[m S S m I I −(m S I )2 ] m O S = g2 m O S +g3m O I

m R S = g2 m R S + g3 m R I M O O = M O O − g1 m O mTO

M O R = M O R − g1 m O mTR M R R = M R R − g1 m R mTR P O O = P O O (25)  where g1 = 1 (m S S + m I I − 2m S I ), g2 = g1 (m I I − m S I ), and g3 = g1 (m S S − m S I ).

(26)

Approximating c1 , c2 , and c3 terms to zero will cause an approximation error for a non-dc frequency. In the frequency band from 0 to ωmax , where ωmax is the highest frequency of interest, the upper bound of the low-pass approximation error for combining a pair of PpI branches can be found as   |m I I − m S I | |m S S − m S I | 2 + (27)  = ωmax pI I − pS I pS S − pS I where the facts that ( p I I − p S I ) and( p S S − p S I ) are all nonnegative real values and all self-inductances are complex number whose imaginary part reflect conductor loss are used. After all the shunt branches are combined and the low-pass approximation is applied, the pth node absorbing process is completed. The resultant potance matrix and inductance matrix are denoted as P( p+1) and M( p+1) , respectively. G. Finding the Most Insignificant Node (MIN) The earlier sections gave the method to absorb the pth nodes assuming that the MIN had been found at the beginning of the pth node absorbing process, as shown in Fig. 2(b). In this section, the method to find the MIN will be derived based on that the MIN is the node that will introduce the minimum low-pass approximation error after the node being absorbed. Therefore, to determine the MIN at the pth iteration, the error bound ( p) (k) of each node k needs to be precomputed for the circuit after absorbing the first ( p−1) nodes. The error bound associated with absorbing each node depends on combining the incremental PpI branches with their respective shunt branches that are PpI branches in the group of “other branches.” Assume the indexes of the incremental PpI branches associated with node k runs from 1 to Nc, and indexes of their shunt PpI branches are numbered according to the index mapping array l in the group of “other branches.” According to (7), (8), and (27), the aggregated error bound for node k can be expressed as  ⎡  ( p)  ( p) NC m I I (i, i ) − m I O (i, l(i ))  ( p) 2 ⎣  (k) = ωmax ( p) ( p) p I I (i, i ) − p I O (i, l(i )) i=1  ⎤  ( p)  ( p) m O O (l(i ), l(i )) − m I O (i, l(i )) ⎦. + ( p) ( p) p O O (l(i ), l(i )) − p I O (i, l(i )) (28)

F. Low-Pass Approximation After combining a pair of shunt PpI branches, the updated elements in (24) contain frequency-dependent terms c1 , c2 ,

According to (16), (18), and (19), the error bound for each node can be precomputed using (5), (6), and (28). The node with the smallest error bound is defined as the MIN.

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Fig. 6. Illustration of internalizing a pseudoinductor by regular inductors. (a) Circuit prior to internalizing a pseudoinductor. (b) Circuit posterior to internalizing the pseudoinductor.

Fig. 5.

as

Flowchart of node absorbing process.

M H. Recursive Node Absorbing Process The node absorbing process is repeated until no node staisfies the preset accuracy criterion δl on the error bound. The whole process can be summarized by three steps as demostrated by the flowchart in Fig. 5.

(i+1)  (i+1) T MC R MR R = (i+1) MC(i+1) MCC R  T  (i+1) T T = v(i+1) vC R  T  (i+1) T T = i(i+1) iC R 

(i+1)

v(i+1) i(i+1)

As said, the pseudoinductors are introduced to retain high fidelity of inductive information in the circuit transformation. After all insignificant nodes are absorbed and the recursive process is stopped, all the pseudoinductors will be internalized by rest of regular inductors in one-by-one fashion. Assuming the first i −1 pseudoinductors have been internalized, the process for internalizing pseudoinductor i is derived here. The circuit with the i th pseudoinductor is shown in Fig. 6(a), in which NC inductors are connected to node i . In the context of internalizing the i th pseudoinductor, the inductors in the circuit can be divided into three groups: 1) the pseudoinductor to be assimilated; 2) the inductors connecting to the pseudoinductor i ; and 3) rest of the pseudo and regular inductors, which are distinguished by subscripts A, C, and R, respectively. The overall inductance matrix of the circuit can be sorted in the form of ⎤ ⎡  (i) T (i) (i) MC R mR A MR R ⎥ ⎢ (i) (29a) M(i) = ⎣ MC(i)R MCC mC(i)A ⎦ .  (i) T  (i) T (i) mR A mC A m AA The corresponding voltage vector and current vector are denotated as T    (i) T T (i) v(i) = v(i) (29b) v v R C A    T  (i) T T (i) i(i) = i(i) (29c) iC iA R respectively. By definition v(i) = j ωM(i) i(i) .

(29d)

Fig. 6(b) shows the circuit with pseudoinductor i being internalized. The corresponding inductance matrix of the resultant circuit, the voltage vector, and the current vector are sorted

(30b) (30c)

respectively. By definition v(i+1) = j ωM(i+1) i(i+1) .

I. Internalization of Pseudoinductors

(30a)

(30d)

According to (30), voltage vC(i+1) can be expressed as (i+1) (i+1) (i+1) + j ωMCC iC . vC(i+1) = j ωMC(i+1) R iR

(31)

Absorbing pseudoinductor i needs to satisfy the following equivalent conditions: jTNC iC(i+1) = i (i) A

vC(i+1) = vC(i) + j NC v (i) A . (i+1)

Therefore, the voltage vC

(32)

can be expressed as

(i+1)

vC

 T (i)  (i) T (i) (i) (i)  = j ωj NC m(i) R A i R + mC A iC + m A A i A  (i) (i) (i) (i)  + j ω MC(i)R i(i) R + MCC iC + mC A i A  (i) T  (i+1)  (i) (i) (i) = j ω MCC + j NC mC A + mC A jTNC + m A A J NC iC  T  (i+1)  iR . + j ω MC(i)R + j NC m(i) (33) RA

Comparing the right-hand sides of (31) and (32) leads to the following updating formulas:  (i) T (i+1) (i) (i) (i) MCC = MCC + j NC mC A + mC A jTNC + m A A J NC   (i) (i) T (i) MC(i+1) M(i+1) (34) R = MC R + j NC m R A R R = MR R . A micromodeling circuit without pseudoinductors can be obtained after all pseudoinductors are internalized. J. Computational Overhead and Parallel Computation In the recursive micromodeling process, the shunt branches combining process dominates computational overhead. As can be found from (25) and (26) that the computations mainly compose of outer products of two vectors in the form of Z = Z + cx · yT

(35)

where Z is an m-by-m matrix, which is a placeholder for inductance matrix M and potance matrix P, x and y are

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m-by-1 vectors, c is a constant, and m represents the number of inductors or potors. The computational overhead of each combining process is in the order of O(m 2 ). Let N be the number of recursive iteration and M be the maximum order of the inductance or potance matrix, the computational overhead of the whole micromodeling process is in the order of O(NM2 ). There are m 2 arithmetic operations for each outer product, which can be computed in parallel using multicores CPU or GPU. It turns out to be a good news because the future trend in developing computing devices will continue to concentrate on multicores rather than the performance of a single thread [36]. That means the micromodeling method can fully utilize the advances on parallel computation, especially on GPUs. To utilize GPU efficiently in developing the micromodeling circuit, two more works need to be done: 1) the amount and frequency of data transactions between CPU and GPU need to be minimized and 2) the amount and frequency of data transactions (read/write) inside GPU need to be minimized. By using the strategies developed recently [37], [38], the proposed micromodeling process can be accelerated by about 30 times as compared to a single CPU process. The numerical examples will demonstrate the superiority of the micromodeling method by adoption of GPU parallel computations.

9

semidefinite matrices. There are some effective enforcement methods for a nonpositive semidefinite matrix [33]–[35]. A simple modified Cholesky algorithm [34] is used, where a generic symmetric matrix F is used as a placeholder for R, M, and P. The eigen decomposition of the N-by-N matrix F can be expressed as F = V VT , = diag(λ1 , . . . , λ N )

(37)

where matrices V and are the eigenvector matrix and eigenvalue diagonal matrix of matrix F, respectively. By adding a corrective matrix F, matrix F is enforced to be a positive semidefinite matrix. The corrective matrix is F = V VT ,  = diag(λ1 , . . . , λ N )

(38)

where the i th diagonal value λi of  is constructed by  0, λi ≥ 0 λi = , i = 1, 2, . . . , N . (39) −λi , λi < 0 The Frobenius norm [32] of F is defined to quantify the passivity violation factor of matrix F   N  2 F F =  λ2i . (40) i=1

IV. PASSIVITY C HECKING AND E NFORCEMENT Unlike the conventional passivity enforcement methods [24]–[30], which are applicable to the state-space matrices of a macro or micromodel but the amended system is not recoverable to a circuit model, the new passivity checking and enforcement method is applicable to a general physically meaningful RLC circuit model, with which while the passivity of the micromodeling circuit is warranted the original circuit configuration is also retained. Let R, M, and P be the resistance matrix, inductance matrix, and potance matrix. According to [31], the cumulative energy at time t of the general RLC circuit can be found as  t 1 1 iTR (τ )Ri R (τ )dτ + iTL (t)Mi L (t)+ qTP (t)Pq P (t) W (t) = 2 2 0 (36) where i R and i L are vectors of currents flowing through the resistors and inductors, respectively; and q P is the vector of charges on the potors. The passivity constraint is derived from the passivity definition that the cumulative energy of a circuit is nonnegative at all times and for all possible excitation signals [13], which leads to Corollary: A general RLC circuit is passive if and only if its resistance matrix R, inductance matrix M, and potance matrix P are all positive semidefinite, or equivalently the eigenvalues of matrices R, M, and P are nonnegative. The corollary not only provides a constraint to check the passivity, but also implies that a nonpassive RLC circuit can be remediated directly by ensuring its resistance matrix R, inductance matrix M, and potance matrix P to be positive

Obviously, the larger the Forbenius norm is, the severer the nonpassivity of the circuit model is. The passivity enforcement scheme is suitable for a physically meaningful micromodeling circuit, whose passivity violation is usually caused by numerical errors and is very small. The numerical examples show that the passivity violation factor of the inductance matrix M of the micromodeling circuit is smaller than that of the method in [12] by three orders of magnitude. V. N UMERICAL E XAMPLES In this section, two examples are used to demonstrate the versatility, scalability, and accuracy of the micromodeling method for high-speed/frequency interconnection and packaging problems. The first example is a low-order PEEC model of an LTCC bandpass filter, which is used to demonstrate the high fidelity of the micromodeling method in preserving the physical essence of the original EM problem. The second example is a multilayer and multiport interconnection circuit with a large-order PEEC model. This example is used to show the scalability and the ability of GPU acceleration of the proposed method. In all numerical examples, the PEEC models are quasi-static and a mixed rectangular and triangular meshing scheme is used. The micromodeling, the PEEC, and the Passive ReducedOrder Interconnect Macromodeling Algorithm (PRIMA) models are executed using single core on a PC with Intel(R) Core(TM) i7-3770 CPU at 3.4 GHz. To demonstrate the suitability of the micromodeling method for GPU acceleration, the GPU module of Nivdia Geforce GTX 980 Ti with 2816 cores is used for demonstration purpose.

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TABLE I C IRCUIT S IZE OF PEEC M ODEL , M ETHOD IN [12], AND T HIS M ETHOD

TABLE II C HANGE OF S YSTEM P OLES D URING R ECURSIVE P ROCESS (U NIT: GHz) Fig. 7. Example of a lossless multilayer LTCC bandpass filter. (a) Circuit layout and meshing scheme. (b) Derived micromodeling circuit.

A. Example 1: LTCC Bandpass Filter The first example is an LTCC multilayer bandpass filter as depicted in Fig. 7(a). The filter consists of three metal layers. The dielectric constant and geometric dimensions are marked in Fig. 7(a). Infinitely thin perfect conductor is assumed. The original PEEC model is generated from the meshing scheme superimposed in Fig. 7(a). Details of the PEEC model are listed in Table I. The order of the PEEC model is reduced by one order of magnitude by both the proposed micromodeling method and the method in [12]. The two micromodeling circuits are compared in the aspects of circuit size, S-parameters, system poles, passivity, and time-domain simulation results. In deriving the proposed micromodeling circuit, the lowpass approximation criterion δl is set to 0.02 and the low-pass cutoff frequency is chosen to be 5 GHz. The micromodeling circuit is superimposed on the filter layout in Fig. 7(b), showing a clear look of its physical meaning. The S-parameters of the two micromodeling circuits are compared with that of the original PEEC model and that of Agilent RF momentum module (ADS) commercial softwave (RF momentum module) in Fig. 8(a) and (b). Both the magnitude and phase of the S-parameters simulated by these methods agree well in the frequency range of intrest from 0.1 to 5 GHz. To make a quasi-static approximation, the multilayer full-wave Green’s function at 10 MHz is used in the PEEC modeling. The system poles of the admittance matrix [16] of the two micromodeling circuits are compared with those of the PEEC model in Table II. The system pole frequencies in some representative steps of the recursive process are listed in ascending order from the most significant frequency (corresponding to the smallest pole) to the least significant frequency. It can be observed that, after absorbing the least important node at a recursive step, the significant system

poles are nearly unchanged and the most insignificant pole is digested and assimilated by the next few least important poles. By comparing the errors of the first three significant system poles of the two micromodeling circuits (i.e., the hatched columns in Table II) with those of the original PEEC model, it can be concluded that the proposed micromodeling circuit introduces smaller errors than those of [12], meaning that physical essence of the PEEC model is better preserved than the existing micromodeling circuit. Passivity violation of the two micromodeling circuits are also compared. The proposed micromodeling circuit does not have any negative self-inductances and the passivity violation factor of its M matrix is 3.9675 × 10−26 . However, the micromodeling circuit in [12] has 23 negative self- inductances and the passivity violation factor of its M matrix is 5.6121 × 10−10 . In this case, the passivity violation factor of M matrix for the original PEEC model is zero. The time-domain responses simulated by the MNA method [4] of the two micromodeling circuits without any passivity enforcement are compared in Fig. 8(c). The input pulse applied at port 1 is with data rate of 1 Gbit/s and rising/falling edges of 0.1 ns. The output signals of the PEEC model and this method match each other very well, while the output signal by the method in [12] does not converge.

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TABLE III C OMPARISON OF C IRCUIT S IZES FOR E XAMPLE 2

Fig. 8. S-parameters obtained by PEEC model, method in [12], this method, and ADS with (a) magnitude and (b) phase. (c) Time-domain response of PEEC model, method in [12], and this method.

Fig. 9. A lossy multiport multilayer PCB circuit. (a) Circuit layout. (b) Mesh details of via holes. (c) Mesh details of differential lines.

B. Example 2: Multilayer Interconnection Circuit To demonstrate the scalability, a typical multiport and multilayer interconnection circuit is considered and is shown in Fig. 9(a). On the circuit board, port 1 and port 2 are the

two pairs of differential input terminals and port 3 and port 4 are the differential output terminals. The overall dimensions of the circuit is 47 244 × 32 512 × 515.62 μm and the dielectric constant is 4.04. The circuit consists of one layer of power plate and two layers of signal traces connected through via holes. The metal thickness is 12 μm and conductivity is set to 5.959 × 107 S · m−1 . Zoomed-in views of via holes and differential signal lines are shown in Fig. 9(b) and (c), respectively. Details of the PEEC model are listed in Table III. Based on the PEEC model, the derived micromodeling circuit contains only 859 nodes as listed in Table III. The low-pass criterion δl is set to 0.03 and the highest frequency of interest is set to 5 GHz. The micromodeling process executed by one core CPU takes 156 min 42 s. The modeling time is reduced to 4 min 46 s by a 2816-core GPU, showing that the proposed method is very suitable for GPU parallel computation. For a reference, if the order of the PEEC model is reduced to the same level by the PRIMA method [16], the PRIMA MOR process costs 286 min 11 s using one CPU core. The simulated S-parameters by the proposed model is compared with those obtained by the original PEEC model, the PRIMA order-reduced model, and the EM simulation of the interconnection circuit layout by ADS, as shown in Fig. 10(a)–(d). The S-parameters of the three models agree well in the frequency range of 0 to 5 GHz. However, the simulation time for the micromodeling circuit and the PEEC model is 4 min 39 s and 1 598 min 13 s, respectively. It is expected that as the size of the original PEEC circuit is reduced by about one order of magnitude, the simulation time for the circuit response is reduced about three orders of magnitude. For reference, the simulation time by ADS, which is exectued by four CPU cores, is 337 min 13 s. Before conducting time-domain simulation, the passivity check needs to be done for the circuit model. It is found that the passivity violation factors for the M matrix of the original PEEC model and the derived micromodeling circuit are 1.8243 × 10−13 and 6.7500 × 10−11 , respectively, although the factors for P and R matrices are zero. The micromodeling circuit derived by the method in [12] contains 592 negative self-inductances, the passivity violation factor of its M matrix is 1.0534 × 10−8 , and the three orders of magnitude larger than that of the circuit model by the proposed

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Fig. 11. Responses of the lossy multilayer multiport interconnection circuit. (a) Responses of a pulse input between ports 1 and 3 simulated by PEEC and the micromodeling circuit. (b) Eye diagram at port 3 by the PEEC model. (c) Eye diagram at port 3 by this micromodeling circuit. (d) Eye diagram at port 4 by the PEEC model. (e) Eye diagram at port 4 by this micromodeling circuit. In eye-diagram simulations, ports 1 and 2 are input ports. TABLE IV C OMPARISON OF C OMPUTATION T IME FOR E XAMPLE 2 Fig. 10. S-parameters obtained by PEEC model, this method, ADS, and PRIMA. (a) Magnitude of S-parameters S11 and S13 . (b) Magnitude of S-parameters S22 and S24 . (c) Phase of S-parameters S11 and S13 . (d) Phase of S-parameters S22 and S24 .

method. The passivity condition of the PEEC model and the proposed micromodeling circuit is enforced with computing time of 522 min 38 s and 16 s, respectively. Having had the passivity of the circuits enforced, a sequence of pulse with data rate of 1 Gbit/s and rising/falling edges of 0.1 ns is applied at port 1 while port 2 is terminated by a matched load. The output responses at ports 3 and 4 are simulated by the MNA method. The time-domain responses at port 3 simulated by the passivity enforced PEEC model and the micromodeling circuit are compared in Fig. 11(a), showing very good agreement. However, as shown in Table IV, simulation time of the micromodeling circuit is three orders of magnitude less than that of the PEEC model. The circuit responses with multiple inputs from multiple ports are straightforward. The advantage of this attribute is obvious for the simulation of eye-diagrams, which usu-

ally involves a long pseudoradom binary sequence (PRBS) with different input situations. When the input signals are applied at ports 1 and 2 using a 256-bit PRBS with the same data rate and the rising/falling edge as the short pulse response for Fig. 11(a), the eye-diagrams of the output responses at ports 3 and 4 are simulated by the passivity enforced PEEC model and the micromodeling circuit as shown

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in Fig. 11(b)–(e). The eye-diagram simulation of the PEEC model costs 14 942 min 49 s (more than ten days). The simulation time is reduced to 34 min 15 s by adopting the micromodeling circuit using the proposed method. VI. C ONCLUSION This paper presents a derived micromodeling circuit for time-domain simulation of a high-speed interconnection and packaging problem. The derivation process of the physically meaningful concise circuit model is straightforward and does not involve any matrix inversion or decomposition. The process is highly suitable for GPU parallel computation. Compared to existing micromodeling methods, by introducing the pseudoinductor to each grounded potor, the physical essence of the original PEEC model can be accurately retained in the circuit transformation. Since the method can reduce the size of a PEEC model by nearly one order of magnitude, the simulation time for both frequency- and time-domain responses can be reduced by about three orders of magnitude. To make the circuit model applicable for time-domain simulation, a pertinent passivity check and enforcement method is also proposed to guarantee the passivity of the derived micromodeling circuit. Two numerical examples are given to demonstrate the versatility, scalability, accuracy, and simplicity of the proposed method through the comparisons of simulation results with the PEEC model, an existing micromodeling circuit method, and the classical Krylov-based PRIMA MOR models. It is expected that the micromodeling circuit will provide a very effective tool for time-domain simulation of high-speed interconnection and packaging problems. R EFERENCES [1] Fundamentals of Signal Integrity, Tektronix, Beaverton, OR, USA, 2009, p. 2. [2] J. Fan, X. Ye, J. Kim, B. Archambeault, and A. Orlandi, “Signal integrity design for high-speed digital circuits: Progress and directions,” IEEE Trans. Electromagn. Compat., vol. 52, no. 2, pp. 392–400, May 2010. [3] A. E. Ruehli, “Equivalent circuit models for three dimensional multiconductor systems,” IEEE Trans. Microw. Theory Techn., vol. MTT-22, no. 3, pp. 216–221, Mar. 1974. [4] C.-W. Ho, A. Ruehli, and P. Brennan, “The modified nodal approach to network analysis,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 6, pp. 504–509, Jun. 1975. [5] A. E. Ruehli and A. C. Cangellaris, “Progress in the methodologies for the electrical modeling of interconnects and electronic packages,” Proc. IEEE, vol. 89, no. 5, pp. 740–771, May 2001. [6] V. Vahrenholt, H.-D. Brüns, and H. Singer, “Fast EMC analysis of systems consisting of PCBs and metallic antenna structures by a hybridization of PEEC and MoM,” IEEE Trans. Electromagn. Compat., vol. 52, no. 4, pp. 962–973, Nov. 2010. [7] L. K. Yeung and K.-L. Wu, “Generalized partial element equivalent circuit (PEEC) modeling with radiation effect,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2377–2384, Oct. 2011. [8] L. K. Yeung and K.-L. Wu, “PEEC modeling of radiation problems for microstrip structures,” IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3648–3655, Jul. 2013. [9] D. Daroui and J. Ekman, “PEEC-based simulations using iterative method and regularization technique for power electronic applications,” IEEE Trans. Electromagn. Compat., vol. 56, no. 6, pp. 1448–1456, Dec. 2014. [10] J. Wang and K.-L. Wu, “A derived physically expressive circuit model for multilayer RF embedded passives,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 5, pp. 1961–1968, May 2006.

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[11] H. Hu, K. Yang, K. L. Wu, and W. Y. Yin, “Quasi-static derived physically expressive circuit model for lossy integrated RF passives,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 8, pp. 1954–1961, Aug. 2008. [12] Y. Dou and K.-L. Wu, “Direct mesh-based model order reduction of PEEC model for quasi-static circuit problems,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 8, pp. 2409–2422, Jul. 2016. [13] S. Grivet-Talocia and B. Gustavsen, “Passivity,” in Passive Macromodeling: Theory and Applications, 1st ed. Hoboken, NJ, USA: Wiley, 2016, pp. 6–13. [14] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 9, no. 4, pp. 352–366, Apr. 1990. [15] P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Padé approximation via the Lanczos process,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 14, no. 5, pp. 639–649, May 1995. [16] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reducedorder interconnect macromodeling algorithm,” IEEE Trans. Comput.Aided Des. Integr. Circuits Syst., vol. 17, no. 8, pp. 645–654, Aug. 1998. [17] F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Guaranteed passive parameterized model order reduction of the partial element equivalent circuit (PEEC) method,” IEEE Trans. Electromagn. Compat., vol. 52, no. 4, pp. 974–984, Apr. 2010. [18] F. Ferranti, G. Antonini, T. Dhaene, L. Knockaert, and A. E. Ruehli, “Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis,” IEEE Trans. Compon., Packag., Manuf. Technol., vol. 1, no. 3, pp. 399–409, Mar. 2011. [19] R. W. Freund, “Model reduction methods based on Krylov subspaces,” Acta Numer., vol. 12, pp. 267–319, Jul. 2013. [20] R. W. Freund, “The SPRIM algorithm for structure-preserving order reduction of general RCL circuits,” in Model Reduction for Circuit Simulation. Dordrecht, The Netherlands: Springer, 2011, pp. 25–52. [21] Y. Su, F. Yang, and X. Zeng, “AMOR: An efficient aggregating based model order reduction method for many-germinal interconnect circuits,” in Proc. Design Autom. Conf., San Francisco, CA, USA, 2012, pp. 295–300. [22] C. S. Amin, M. H. Chowdhury, and Y. I. Ismail, “Realizable reduction of interconnect circuits including self and mutual inductances,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 24, no. 2, pp. 271–275, Feb. 2005. [23] Z. Ye, D. Vasilyev, Z. Zhu, and J. R. Phillips, “Sparse implicit projection (SIP) for reduction of general many-terminal networks,” Design Autom. Conf., San Francisco, CA, USA, 2008, pp. 1–2. [24] R. N. Shorten, P. Curran, K. Wulff, and E. Zeheb, “A note on spectral conditions for positive realness of transfer function matrices,” IEEE Trans. Autom. Control, vol. 53, no. 5, pp. 1258–1261, Jun. 2008. [25] S. Boyd, V. Balakrishnan, and P. Kabamba, “A bisection method for computing the H∞ norm of a transfer matrix and related problems,” Math. Control Signals Syst., vol. 2, no. 1, pp. 207–219, Jan. 1989. [26] S. Grivet-Talocia, “Passivity enforcement via perturbation of Hamiltonian matrices,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 9, pp. 1755–1769, Sep. 2004. [27] Z. Bai and R. W. Freund, “Eigenvalue-based characterization and test for positive realness of scalar transfer functions,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2396–2402, Dec. 2000. [28] C. Schroeder and T. Stykel, Passivity Check in Passivation of LTI Systems, document TR-368-2007, 2007. [29] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance matrices approximated by rational functions,” IEEE Trans. Power Syst., vol. 16, no. 1, pp. 97–104, Feb. 2001. [30] W. D. C. Boaventura, A. Semlyen, M. R. Iravani, and A. Lopes, “Sparse network equivalent based on time-domain fitting,” IEEE Trans. Power Del., vol. 17, no. 1, pp. 182–189, Jan. 2002. [31] C. K. Alexander and M. N. O. Sadiku, “Magnetically coupled circuits,” in Fundamentals of Electric Circuits, New York, NY, USA: McGraw-Hill, 2007, pp. 564–566. [32] G. H. Golub and C. F. van Loan, “Positive definite system,” in Matrix Computation, 4th ed. Baltimore, MD, USA: The Johns Hopkins Univ. Press, 2013, p. 160. [33] N. J. Higham, “Computing the nearest correlation matrix—A problem from finance,” IMA J. Numer. Anal., vol. 22, no. 3, pp. 329–343, 2002. [34] S. H. Cheng and N. J. Higham, “A modified Cholesky algorithm based on a symmetric indefinite factorization,” SIAM J. Matrix Anal. Appl., vol. 19, no. 4, pp. 1097–1110, Oct. 1998. [35] N. J. Higham, “Computing a nearest symmetric positive semidefinite matrix,” Linear Algebra Appl., vol. 103, pp. 103–118, May 1988.

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[36] J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU computing,” Proc. IEEE, vol. 96, no. 5, pp. 879–899, May 2008. [37] Y. Dou and K.-L. Wu, “Acceleration of physically derived micromodeling circuit for packaging problems using graphics processing units,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 1638–1640. [38] Y. Dou and K.-L. Wu, “Acceleration of parallel computation for derived micro-modeling circuit by exploiting GPU memory bandwidth limit,” in Proc. IEEE Int. Conf. NEMO, Seville, Spain, May 2017, pp. 146–148.

Yuhang Dou (S’13) received the B.S. degree in electronic engineering from the Nanjing University of Science and Technology, Nanjing, China, in 2012. She is currently pursuing the Ph.D. degree at the Chinese University of Hong Kong, Hong Kong. Her current research interests include partial element equivalent circuit, full-wave circuit domain modeling for signal integrity and electromagnetic compatibility problems, and physics-based modelorder reduction of PEEC model for electromagnetic problems in both the frequency and time domains.

Ke-Li Wu (M’90–SM’96–F’11) received the B.S. and M.Eng. degrees from the Nanjing University of Science and Technology, Nanjing, China, in 1982 and 1985, respectively, and the Ph.D. degree from Laval University, Quebec, QC, Canada, in 1989. From 1989 to 1993, he was with the Communications Research Laboratory, McMaster University, Hamilton, ON, USA, as a Research Engineer and a Group Manager. In 1993, he joined the Corporate Research and Development Division, COM DEV International, as the largest Canadian space equipment manufacturer, where he was also a Principal Member of Technical Staff. Since1999, he has been with the Chinese University of Hong Kong, Hong Kong, where he is a Professor and Director of the Radio Frequency Radiation Research Laboratory. He has authored or co-authored numerous publications in the areas of EM modeling and microwave passive components, and microwave filter and antenna engineering. His current research interests include EM-based circuit domain modeling of high-speed circuits, microwave passive circuits and systems, synthesis theory and practices of microwave filters, antennas for wireless terminals, LTCC-based multichip modules, and RF identification technologies. Prof. Wu is a member of the IEEE MTT-8 Subcommittee (Filters and Passive Components) and also serves as a TPC member for many prestigious international conferences including International Microwave Symposium. He was an Associate Editor of the IEEE MTT-S IEEE T RANSACTIONS ON M ICROWAVE T HEORY T ECHNOLOGY from 2006 to 2009. He was a recipient of the 1998 COM DEV Achievement Award for the development of exact EM design software of microwave filters and multiplexers and Asia–Pacific Microwave Conference Prize in 2008 and 2012.

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Single-Band and Switchable Dual-/Single-Band Tunable BPFs With Predefined Tuning Range, Bandwidth, and Selectivity Di Lu , Student Member, IEEE, Xiaohong Tang, Member, IEEE, N. Scott Barker, Senior Member, IEEE, and Yukang Feng Abstract— This paper presents a new kind of highly flexible frequency-agile bandpass filters (FA-BPFs) based on the novel synchronously tuned dual-mode resonator (STDR). The bandwidth (BW), BW variation tendency, passband selectivity (stopband rejection level), and frequency tuning range of the filter can be predefined individually. Benefiting from the unique characteristics of the STDR, the FA-BPF with very simple and highly flexible design/control procedures is achieved. Due to the proposed geometry, two self-adaptive transmission zeros are introduced and move with the passband. The predefined mechanism is investigated in detail, and the simple design and predefined procedures are summarized. Then, three 0.75–1.7 GHz single-band examples with elliptic response are developed to achieve three predefined absolute BWs (ABW). The design techniques and filter superiority are confirmed by the experiments. Moreover, aiming at China 2G/3G/4G cellular wireless/mobile communication system (up to band 40:0.825–2.65 GHz), a novel intrinsically switchable single-/dual-band FA-BPF is presented based on the proposed STDR. An example with a constant ABW1 dB and a fractional bandwidth1 dB (FBW1 dB ) is designed to validate the theory and analysis. The FA-BPF is able to operate as a highly selective dual-band FA-BPF with 0.76–1.78 GHz/1.61–2.63 GHz tuning ranges, and also can be switched to single-band operation with the continuous tuning range of 0.76–2.63 GHz. Index Terms— Predefined bandwidth (BW), predefined stopband, predefined tuning range, switchable single-/dual-band FA-BPF, 2G/3G/4G mobile communications frequency-agile bandpass filters (FA-BPFs).

I. I NTRODUCTION N RECENT decades, electrically tunable planar filters have received increasingly great attention from both the academic world and industry field, due to their ability to

I

Manuscript received May 5, 2017; revised July 14, 2017 and September 12, 2017; accepted October 16, 2017. Date of publication November 27, 2017; date of current version March 5, 2018. This work was supported by the National Natural Science Foundation of China under Grant 61701052. (Corresponding author: Di Lu; Xiaohong Tang.) D. Lu is with the EHF Key Laboratory of Science, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China, and also with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904 USA (e-mail: [email protected]; [email protected]). X. Tang is with the EHF Key Laboratory of Science, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China. N. S. Barker and Y. Feng are with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2772816

substantially reduce size for the multiband transceiver front end and because of their high flexibility for cognitive/softwaredefined radio [1]–[3]. During this period, a large number of high-performance tunable filters were reported to provide possible solutions. However, achieving a broad frequency covering range (50%) is still a challenge for tunable filter design, especially with a constant bandwidth (BW). On the other hand, the emerging carrier aggregation system allows a total BW of up to 100-MHz mobile broadband wireless communications to meet the IMT-advanced requirements [4]–[6]. Such a great expectation places an urgent demand on the wide tuning range dual-band tunable components. Unfortunately, the dual-band tunable filter with a broad tuning range as a promising solution has seldom been reported. This is because the sufficient stopband BW for the second passband, and wide tuning range for two passband are not easy to be achieved simultaneously. For single-band frequency-agile bandpass filters (FA-BPF), several approaches were frequently utilized to handle such a wide frequency tuning range (≥50%) and achieve the constant BW response. The first approach is directly arranging different frequency filters in parallel with channel switches to form the switchable filter banks, such as stepped-impedance resonator -based coupling filter bank [7] and BPF-LPF filter bank [8]. However, these designs required a very large space due to the duplication of filters for each channel. The second approach is elaborately controlling the electric and magnetic coupling between the resonators to realize a relatively stable coupling coefficient over the tuning range, thus resulting in an FA-BPF with constant BW [9]–[14]. The tunable passband with constant BW was obtained by adopting the reported structures, whereas very few of them had a frequency tuning range over an octave (50%) and their selectivity was poor, especially for the highly practical two-pole BPF. This is due to the fact that the mixed coupling technology hindered the self-adaptive TZ generation. The third category is adding the extra variable capacitors to control the coupling directly, thus alleviating the need for the stable coupling structures [15]–[18]. High selectivity could be achieved, but both design procedure and control mechanism were much more complex, because of the extra capacitors and bias. Additionally, adopting tunable multimode resonators to construct an FA-BPF can be classified as the fourth approach. Benefiting from the property that the multiple modes coexist in one resonator without the mutual coupling, the circuit size and design complexity for a given degree

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of the FA-BPF are reduced. Dual-mode tunable resonators [19] and tri-mode tunable resonators [20] were employed to implement 0.6–1 and 0.6–1.45 GHz frequency agility with constant BW, respectively. However, narrow tuning range [19], complex design, and control procedures [20] limited their development. Aside from the conventional FA-BPF discussed above, three basic approaches were reported to extend the frequency tuning range dramatically. In [21], three standalone two-pole FA-BPFs with 50% tuning range were stacked to cover a 144.4% frequency (0.1–0.62 GHz) tuning range. However, it suffered from bulky volume and the extra switch pairs. In [22], dual zero-value coupling technique were applied into pseudo dual-band FA-BPFs (not a dual-band tunable filter), and the widest tuning range of 0.75–0.99 and 1.05–1.44 GHz could be switched ON and OFF. However, the filter tuning range was segmented, and extra varactors complicated the tuning mechanism and design process. More recently, a continuously tuning FA-BPF with a constant BW covering 0.55–1.9 GHz was reported in [23]. The frequency tuning range could be switched by p-i-n diodes between 0.55–0.99 and 0.99–1.9 GHz, while the resonating structure was switched between two varactor-loaded λ/2 resonators and one varactor-loaded λ/4 resonator. The widest tuning range was implemented; however, the design process was complicated and the selectivity needed to be improved by introducing the transmission zeros. For dual-band FA-BPF, a broad tuning range with high selectivity as well as a constant BW is preferable. However, few reported dual-band FA-BPFs achieved these characteristics simultaneously. In [24], a 0.77–1 GHz/1.57–2 GHz dual-band FA-BPF with a single-band constant BW; in [25], a 1.48–1.8 GHz/2.40–2.88 GHz filter without constant BW; in [26], a 1.15–1.6 GHz/2.12–2.45 GHz filter with one constant BW; in [27], a 0.98–1.22 GHz/1.63–1.95 GHz filter with two constant absolute BW (ABW); in [28], a 1.85–2.67 GHz/3.84–5.34 GHz tunable filter using halfsubstrate integrated waveguide without constant BW; and in [29], a 0.617–0.817 GHz/1.386–1.68 GHz lumped dualband filter with one constant BW were reported. It is found that the tuning range of the reported dual-band FA-BPFs was difficult to cover 50% and their selectivity still needed to be improved (e.g., to achieve elliptic response). The dualband tunable elliptic response can be acquired by utilizing the quasi-BPF configuration with nonresonating nodes [30]–[32]. However, the frequency tuning range and their stopband BW were compromised. Bandpass-to-bandstop switchable filters with single and dual passband were reported in [33], which exhibited many application perspectives because of its flexibility. But their frequency covering range were limited. In this paper, for the sake of providing a solution for cognitive/software-defined radio and addressing aforementioned issues, a simple synchronously tuned dual-mode resonator (STDR) is proposed to achieve a highly selective FA-BPF configuration with the predefined ABW and frequency tuning range. Two resonant modes of STDR can be tuned synchronously by varying only one bias, and their separation and frequency range can be simply controlled by

Fig. 1. Proposed STDR. (a) Transmission line model. (b) Even-mode equivalent circuit. (b) Odd-mode equivalent circuit.

changing varactor-loading positions, shorted stub length, or the stepped-impedance ratio. The design formulas are established, and the predefine mechanism of the proposed STDR are investigated statistically. Based on the proposed resonator, a new highly selective filter is presented. In addition to the BW and tuning range predefine, the proposed filter structure is capable of predefining the passband selectivity (or stopband rejection) by preseting two transmission zeros, and individually predefining the external Q e to obtain the very good impedance matching over the entire frequency tuning range. For demonstration, three 0.76–1.78 GHz FA-BPFs are designed to achieve such a broad frequency covering range and realize different ABW variations: constant ABW (C.ABW), increasing ABW (I.ABW), and decreasing ABW (D.ABW). Finally, to offer a solution for the Carrier Aggregation Systems and cover entire 2G/3G/4G spectrum allocation for IMT in China (0.825–2.65 GHz), a highly selective switchable dual-/single-band FA-BPF technique is proposed. The filter design theory and techniques are investigated in detail. A demonstrative filter are engineered to cover 0.76–1.78 GHz with C.ABW for the first band and 1.61–2.63 GHz with constant fractional BW (FBW) for the second band. When it switches to the single-band FA-BPF, the filter is able to be continuously tuned from 0.76 to 2.63 GHz. To the best of the authors’ knowledge, a dual-band FA-BPF with nearly constant BW covering the entire 2G/3G/4G spectrum is achieved for the first time. II. BASIC T HEORY AND T ECHNOLOGY A. Synchronously Tuned Dual-Mode Resonator Fig. 1(a) presents the proposed STDR, where an E-shape stepped-impedance dual-mode resonator with low-admittance sections (Y1 , θ1 ), high-admittance sections (Y2 , θ2 ), and shorted stub section (Ys , θs ), is controlled by only two variable capacitors. The capacitors are loaded at the low admittance sections with an electrical length of θt from the center plane. Considering the symmetrical structure, the even-odd mode analysis method is taken into account by adding the magnetic wall and electric wall to facilitate the analysis. The corresponding circuits are given in Fig. 1(b) and (c). Thus, the even-mode and odd-mode admittance can be calculated by Yinc2 = ( j Y2 tan θ2 ) Yinc2 + j Y1 tan(θ1 − θt ) Yinc1 = Y1 Y1 + j Yinc2 tan(θ1 − θt )

(1) (2)

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Fig. 2. Typical response of the STDR using the week couple excitation and the resonant mode variation by tuning Ct form 0.69 to 14 pF.

Yinc = Yine2 = Yine1 = Yino1 = Yine =

Yinc1 j ωCt Yinc1 + j ωCt (− j Ys cot θs )/2 Yine2 + j Y1 tan θt Y1 Y1 + j Yine2 tan θt (− j Y1 cot θt ) Yine1 + Yinc

Yino = Yino1 + Yinc

(3) (4) (5) (6) (7) (8)

where the reference frequency is f d . Under the resonant condition (Im{Yine/ino} = 0), the resonant mode frequencies are extracted by calculating (1)–(8). To investigate the STDR, its typical response needs to be checked first, which is obtained by using the weak coupling excitation method [34]. As shown in Fig. 2, the two resonant modes are tuned simultaneously by changing Ct , and the separation between them is notably stable as the frequency increases. Synchronism of two resonant modes can be observed in the inset of Fig. 2. As a result, the proposed STDR is highly suitable for FA-BPF designs. Assuming the center frequency equal to the arithmetic average of the two resonant modes, f cen = ( fo + f e )/2, and the BW equal to the separation, f BW = f o − f e , the proposed STDR can be quantitively characterized by solving the equation Im{Yine/ino } = 0. Additionally, the commercial varactor model, SMV1281 with a tuning range from Ctmin = 0.69 pF to Ctmax = 14 pF is chosen as the tuning device. Thus, the tuning range can be defined as α=

[ f o (Ctmin ) − f e (Ctmin )] − [ f o (Ctmax ) − f e (Ctmax )] . {[ f o (Ctmin ) − f e (Ctmin )] + [ f o (Ctmax ) − f e (Ctmax )]}/2 (9)

When stepped-impedance ratio k = Y2 /Y1 is set, the relationship between k and α can be obtained, as shown in Fig. 3(a). For given parameters, when the larger stepped-impedance ratio is chosen the wider frequency tuning range can be achieved. It is seen in Fig. 3, using uniform impedance STDR, the tuning range is about 0.6, which has been reported in [35]. In this paper, stepped-impedance ratio k = 3 is utilized to extend the frequency tuning range to about 0.73. Thus, the frequency tuning range can be approximately predefined by choosing the appropriate k. Comparing with the uniform impedance STDR, the steppedimpedance STDR is not only used to control the frequency tuning range but also to reduce the total size of the resonator.

Fig. 3. (a) Frequency tuning range versus stepped-impedance ratio k. (b) Size reduction ratio versus stepped-impedance ratio k. (c) Fundamental/spurious passband frequency ratio versus tuning capacitor with different k.

Fig. 3(b) presents the size reduction ratio versus the steppedimpedance ratio k, where the size reduction ratio is defined as the length ratio between stepped-impedance STDR and uniform impedance STDR. As shown, the larger the steppedimpedance ratio, the smaller is the STDR size. When steppedimpedance ratio is set as 3, the length of STDR is reduced to 77%. Stopband performance is also important for the exploitation of wide tuning range or multiband or multichannel filter applications. For planar filters, the stopband performance is mainly determined by the spurious passband location. To investigate the spurious-band performance, the center frequency of the spurious passband is assumed as the arithmetic average of the two harmonic resonant modes f cen2 = ( f o2 + fe2 )/2, where fo2 and fe2 are the first harmonics of the odd- and even-mode resonant frequencies. Thus the frequency ratio of fundamental/spurious passbands can be defined as f o2 (C) + f e2 (C) f cen2 . (10) = f cen f o (C) + f e (C) Fig. 3(c) shows the fundamental/spurious passband frequency ratio with the same parameters given in Fig. 3(a). As shown, the frequency ratio β changes as the frequency is tuned by tuning the capacitor. However, the frequency ratio β is dominated by k. When k is adjusted from 1 (uniform impedance STDR) β=

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Fig. 5. Schematic layout and coupling topology of the single-band FA-BPFs for (a) C.ABW and I.ABW, and (b) D.ABW. (c) Coupling topology and STDR dimensions. TABLE I E LECTRICAL PARAMETERS OF THE U SED STDR FOR T HREE C ASES

Fig. 4. Ct versus f BW and f cen . (a) With different θt where θs = 9°. (b) With different θs where θt = 0.5°.

to 3 (used in this paper), spurious-band free space (upper stopband) increases by almost one third, which provides sufficient space for wide tuning range and multichannel filter applications. Uniquely, the separation between the two modes is controllable. To investigate this feature, the following dimensions f d = 1.2 GHz, k = 3, Y2 = 1/25 S, Ys = 1/75 S, θ1 = 53°, θ2 = 46° are used and the initial values are set as θs = 9° and θt = 7°, thus allowing the fcen covering 0.75–1.7 GHz. Fig. 4(a) and (b) shows the f BW and f cen variations with different θs and θt by tuning the capacitor Ct . As shown in Fig. 4(a), changing θt adjusts the separation of two resonances f BW in the high-frequency area (small Ct ) with almost no influence on the entire fcen curve or f BW in the low-frequency area (large Ct ). When θt is set as θt = 7°, an approximately constant frequency separation f BW (Ct = 0.69) = f BW (Ct = 14) ≈ 120 MHz are achieved. Thus, when the condition θt > 7° or θt < 7° are given, an increasing fBW or decreasing f BW can be obtained, respectively. In addition, as shown in Fig. 4(b), adjusting the length of the shorted stub θs affects the frequency separation f BW location. It is seen that with the variation of θs , the tuning range of fBW is relocated while the f cen curve and f BW variation tendency are almost fixed. When θs varies from 7° to 15°, the f BW independently changes from narrow BW (small f BW ) to wide BW (large f BW ) with a nearly constant curve shape. Therefore, the variation tendency of two resonant modes and BW range location can be independently predefined by choosing appropriate θt and θs . B. External Quality Factor Q e and Stopband Predefined Technologies Table I lists the electrical parameters of three demonstrative STDRs which are engineered to achieve C.ABW, I.ABW, and D.ABW. Two quite useful filter layouts are given in

Fig. 5(a) and (b), while their corresponding coupling topology and physical dimensions are shown in Fig. 5(c). The substrate Rogers RT/duroid 5880 (h = 0.508 mm, εr = 2.2, tan δ = 0.0009) and Skyworks SMV1281-079LF (L s = 0.7 nH, Rs = 1.7 , Ct = 0.69–14 pF) are adopted to demonstrate the external quality factor Q e and stopband predefined technologies. For analysis and design convenience, the external quality factor of the filter is defined as the arithmetic average of evenmode and odd-mode external quality factors Q e = (Q exe + Q exo )/2. Thus, with the given parameters, Q e curves for filter 1 [Fig. 5(a)] can be extracted, as shown in Fig. 6. Apparently, when w0 is adjusted from 0.1 to 3 mm, the Q e curve changes from increase to decrease, and when the st is adjusted from 0.05 to 0.2 mm, the Q e curve moves from small value area to large with the nearly constant shape. For filter 2 [Fig. 5(b)], it shows the same characteristic which is not repeatedly discussed here. Therefore, it is concluded that the slope and the value of Q e are predefined by choosing appropriate w0 and st . Comparing with two feeding structures, the second one [Fig. 5(b)] provides the stronger coupling leading to the smaller Q e . In the proposed filter structures, source-load coupling which is controlled by wio or gio affects the stopband performance a lot. Fig. 7 presents the typical responses when only wio or gio is adjusted. Obviously, source-load coupling introduces two transmission zeros at both sides of the passband. By increasing wio or decreasing gio , the two transmission zeros move toward passband resulting in the higher passband selectivity and compromised stopband rejection level. As a conclusion, the designer can choose wio or gio to predefine the desired stopband and selectivity performance without affecting the in-band performance.

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f exe/exo π f e/o τe/o = Q exe/exo =

f ±90 2  −1 S21 = 2(1/Q e )[ A] N,1 S11 = 1 − 2(1/Q e )[ A]−1 1,1 .

(14) (15)

Therefore, the complete design procedure for this type of FA-BPFs is summarized as follows. 1) According to the prescribed center frequency, ripple level, frequency tuning range, BW, and BW variation, extract the element-variable coupling matrix [36], [37]. 2) Specify the electrical parameters according to (1)–(8) and physical dimensions by EM design to approach the required coupling element curves. (Specify k to set the tuning range, specify θt to set the slope of BW variation, and specify θs to set the BW.) 3) Add feeding structure to implement the filter. Optimize w0 and st to approach the prescribed Q e curves (filter ripple level), and optimize wio or gio to approach the desired stopband.

Fig. 6. Extratced Q e curves by (a) adjusting w0 with st = 0.12 mm and (b) adjusting st with w0 = 1.5 mm.

Fig. 7. Typical passband responses by adjusting wio = 1/2/4 mm for (a) C.ABW filter and (b) I.ABW filter, or gio = 0.05/0.2/1 mm for (c) D.ABW.

C. Design and Predefined Procedure According to [36], a dual-mode tunable filter can be represented by a denormalized element-variable coupling matrix as     (Q exe + Q exo ) x 0 m ee m eo m = = Qe = 0 y m oe m oo 2 (11) (12) x = m ee = ( f e / f d − f d / f e ) y = m oo = ( f o / f d − f d / f o )

(13)

III. S INGLE -BAND FA-BPF S W ITH S ELECTIVITY P REDEFINE A. Design In this section, three examples are designed with C.ABW, I.ABW, and D.ABW. Three FA-BPFs utilize the same STDR with different shorted stub and capacitor loading positions as discussed before. The feeding structure 1 [Fig. 5(a)] is adopted for C.ABW and I.ABW filter design, while the feeding structure 2 [Fig. 5(b)] is adopted for D.ABW filter because of its stronger coupling requirement. The substrate Rogers RT/duroid 5880 and SMV1281-079LF are adopted for the designs, while the full-wave EM simulator ANSYS HFSS and circuit simulator Keysight ADS are employed for the physical dimension optimization. The target performance of the filters are summarized as follows: 1) f cen : 0.75–1.7 GHz; 2) BW1 dB : 85 MHz/100–300 MHz/130–80 MHz for C.ABW/I.ABW/D.ABW. To achieve the desired ABW variation tendency in the three cases, the function curves of Q e are required to be specified by optimizing w0 for the slope of Q e , and optimizing st for the values of Q e . It is worth to be noted that the feeding structure optimization, is used not only to control the coupling strength (Q e ), but also to help to stabilize the BW variation. For instance, for I.ABW filter design, it can help to enlarge the BW at the lower frequency area (large Ct ) and reduce the BW at the upper-frequency area (small Ct ). According to [36], a frequency-fixed filter is considered for the three cases with the following specifications. Chebyshev: f d = 1.2 GHz

FBW = 8.75%

RL = 15 dB.

(16)

The associated coupling matrix can be extracted using the method described in [37] as x = 0.1126

y = −0.1126

Q e = 21.26.

(17)

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Desired and extracted Q e for the three FA-BPFs.

Thus, for the dual-mode tunable filter, three possible coupling matrices (18)–(20) with variable elements are obtained as the design target. C.ABW: 1.024 > x > −0.6

0.7 > y > −0.75

14.11 < Q e < 30.4. (18)

I.ABW: 1.024 > x > −0.543

0.7 > y > −0.95

30.4 < Q e < 11.83.

(19)

D.ABW: 1.04 > x > −0.525 10.56 < Q e < 33.6.

0.7 > y > −0.626 (20)

Three calculated Q e curves, as the target of the EM optimization process, are depicted in Fig. 8. With these design parameters, I/O feeding structures of the three filters can be designed by virtue of the full-wave simulation. The final Q e curves are compared with the calculated ones, as shown in Fig. 8. The associated dimensions are w0 = 0.45 mm, st = 0.12 mm for Fig. 5(a) and w0 = 0.2 mm, st = 0.1 mm for Fig. 5(b). After the EM design process for taking into account the parasitic effects, all the dimensions are determined and passband responses are specified. Following the design procedure, at this point, the passband selectivity (or stopband rejection level) can be predefined by adjusting the source-load coupling structures (i.e., wio for C.ABW filter and I.ABW filter, and gio for D.ABW filter) as discussed before. wio = 2 mm and gio = 0.2 mm are used, and the demonstrative circuits are fabricated accordingly. The photographs for three examples and their key dimensions are presented in Fig. 9(a)–(c). B. Measurement Three fabricated FA-BPFs are measured by the National Instruments’ vector network analyzer, while a 0–25 V dc bias is fed to the varactors through the RF block resistors R = 10 k . The measurement and simulation results are shown in Fig. 10(a)–(c). As can be seen, the center frequency of the C.ABW FA-BPF is tuned from 0.76 to 1.78 GHz

Fig. 9. Photographs of the three FA-BPFs with (a) C.ABW, (b) I.ABW, and (c) D.ABW.

with a highly selective passband, and it exhibits an approximate C.ABW1 dB 84 ± 14 MHz. For the I.ABW FA-BPF and D.ABW FA-BPF, the center-frequency tuning range are 0.75–1.87 and 0.74–1.69 GHz, respectively, while the ABWs are changed from 75 to 285 MHz and from 151 to 82 MHz. The highly selective passband is maintained due to two selfadaptive transmission zeros. Note that, for D.ABW FA-BPF [Fig. 10(c)], the extra transmission zero ( f z3 ), introduced by the terminated parallel-coupled line [38], further strengthens the upper stopband rejection. Fig. 11 summarizes the ABW and insertion loss performance. The ABW1 dB variation tendencies of the three filters agree well with simulations, and the insertion losses are 4.5–2, 4.2–1.2, and 2.8–1.8 dB, respectively, for C.ABW, I.ABW, and D.ABW. As verified in [39], the 1-dB compression point (P1 dB ) is not able to truly represent the large-signal S-parameters in tunable filters since the response shifts in frequency at lower power levels. Therefore, the responses of the three FA-BPFs with the different input power levels are measured. As shown in Fig. 12, the three filters can handle at least 13-dBm input signal without a considerable distortion over the tuning range. The input third-order intercept points (IIP3s) of the filters, using 2-MHz spacing two tones, are measured following the method [35]. As shown in Fig. 13, the measurement results are 11–20 dBm IIP3 for C.ABW filter, 15.5–25.5 dBm IIP3 for I.ABW filter and 18.5–23 dBm IIP3 for decreasing filter. Table II lists the performance comparisons between the three prototype filters and the other high-performance tunable filters.

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TABLE II P ERFORMANCE C OMPARISONS OF M EASURED R ESULTS W ITH O THER T UNABLE F ILTERS

Fig. 11.

Measured insertion loss and ABW1 dB versus frequency.

Fig. 12.

Measured responses with different power levels of input signal.

elements and one bias are needed. The ability to simultaneously predefine the ABW, stopband performance, Q e and frequency tuning range of a tunable filter ABW is achieved for the first time. The relatively larger BW deviation for the C.ABW BPF is mainly because of the wide frequency tuning range [19]. Fig. 10. Measurement and simulation results of the three demonstration FA-BPFs with (a) 0.76–1.78 GHz f cen and C.ABW1 dB 84 ± 14 MHz, (b) 0.75–1.87 GHz f cen and I.ABW1 dB from 75 to 285 MHz, and (c) 0.74–1.69 GHz f cen D.ABW1 dB from 151 to 82 MHz.

IV. I NTRINSICALLY S WITCHABLE D UAL -/S INGLE -BAND FA-BPF W ITH C ONSTANT ABW AND FBW A. Design Theory and Analysis

The broad frequency tuning ranges with highly selective tunable passbands are highlighted. The straightforward filter design/control procedures are obtained since only two tuning

Coupling topology of the switchable dual-/single-band FA-BPF is proposed in Fig. 14. Two coupling channels with two STDRs constitute two tunable passbands, and source/load

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Fig. 16.

Switch-ON and -OFF states of the C.ABW FA-BPF.

Fig. 13. Measured IIP3 of the three FA-BPFs with 2-MHz spaced frequency two tones.

Fig. 14. Coupling topology of the intrinsically switchable dual-/singleband FA-BPF. (a) Dual-band operation. (b) Single-upper band operation. (c) Single-lower band operation.

Fig. 15. Extracted Q e curves of the proposed C.ABW FA-BPF in Fig. 9(a) with and without loading another STDR.

coupling generates four self-adaptive transmission zeros to form the steep passband skirts. When the filter works as a single-band FA-BPF, as shown in Fig. 14(b) and (c), one passband is switched OFF, and the other is maintained. Since the dual-band operation works at different frequencies and the parallel-coupled line feeding topology are adopted, the loading effects of one channel to the other channel are negligible. In order to demonstrate this negligible loading effect, another high-frequency STDR is loaded on C.ABW FA-BPF (insets of Fig. 15), and thus the Q e curves with and without the loaded STDR can be extracted. Fig. 15 presents two extracted Q e curves. As shown, there is only a small discrepancy between them in the high-frequency area, which is negligible for tunable filter design. Besides, For this case, the loading effect further linearizes the Q e curve of the filter which results in the better return loss over the tuning range. Based on the analysis above, each frequency channel is independent and can be separately designed without considering

the loading effect. Thus, the design theory is different from the design topology in [40]. Switch-OFF state of the proposed filters (Fig. 9) is needed to be exploited to enable the single band operation for the dualband filter. Aside from the simple one-bias control, applying different voltages to two varactors in an STDR leads to the impedance mismatch. Taking the proposed C.ABW FA-BPF as an example (Fig. 16), it can be found when 0/25 V voltages apply to the varactors, the passband collapses with −20 dB S12 . The corresponding Smith chart shows that the passband frequency shifts from the center point to the edge of the circle which indicates that the filter switches from match to mismatch. At this point, the analysis above has confirmed the realizability of the coupling topology. Independently designing, tuning and switching OFF the each passband of the filter are all available. To cover entire 2G/3G/4G spectrum allocation for IMT in China (0.825–2.65 GHz), the C.ABW FA-BPF [Fig. 9(a)] has been considered as the lower band channel. Then, the remaining frequency range is 1.78–2.65 GHz (35.2%), which can be easily covered by adopting the uniform impedance STDR [35]. To further exhibit the flexibility of the STDR and simplify the design procedure, the uniform impedance STDR with constant FBW is considered to achieve the upper channel. Besides, the highest frequency of the second passband is arranged far away from the spurious band to avoid the spur loading effects. B. Design The proposed structure of switchable dual-/single-band FA-BPF is depicted in Fig. 17(a). The desired performance of the ideal dual-/single-band FA-BPF is specified as follows: 1) f cen : 0.8–1.75 GHz/1.7–2.65 GHz; 2) BW1 dB : 84 MHz/5%. According to [36], the filter prototype with the specifications (21) is used for the upper band design. The element-fixed matrix (22) derived by the method [37] is employed. The Q e is fixed, because of the constant FBW. Butterworth: fd = 2.1 GHz

FBW = 5.5%

0.467 > x > −0.43

RL = 15 dB

0.39 > x > −0.508

(21)

Q e = 43.25. (22)

As discussed before, uniform impedance STDR is adopted to achieve the upper band as shown in Fig. 17(b). Following

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Fig. 19.

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Photographs of dual/single switchable FA-BPF.

Fig. 17. Switchable dual-/single-band FA-BPF. (a) Schematic layout. (b) Constant FBW STDR with specifications. (c) FBW and center-frequency response of the FBW STDR.

Fig. 18.

Desired and extracted Q e for the upper band.

the design method given in [35], the electrical parameters of the desired STDR are obtained and presented in Fig. 17(b) as well. The αFBW and f cen curves as functions of Ct are calculated and plotted in Fig. 17(c), where αFBW = fcen / f bw is utilized to predefine the constant FBW1 dB . As shown, with changing Ct from 0.69 to 14 pF, the center frequency varies from 2.65 to 1.69 GHz while the αFBW is kept nearly a constant (0.0389–0.0395). With the calculated electrical parameters of the uniform STDR, the filter dimensions can be obtained by the EM design according to the design procedure, and the desired Q e can be approximately achieved by optimizing stb and w0 . The final extracted Q e and required Q e curves with the associated physical dimensions are presented in Fig. 18. The reasonable agreement between two curves is acceptable for the tunable filter design. Additionally, Fig. 19 also presents the extracted Q e curve with loading effect. It is seen, the loading effect makes a little impact on the Q e curve in the low-frequency area, but this impact is negligible as well.

Fig. 20. Measurement and simulation results of the dual-band FA-BPF. (a) Tuning the first band from 0.76 to 1.78 GHz with fixed second band at 2.63 GHz and C.ABW 79 ± 9 MHz. (b) Tuning the second band from 1.6 to 2.63 GHz with fixed first band at 0.76 GHz and constant FBW 5.05% ± 0.25%.

Thus, all dimensions are determined, and the corresponding filter is fabricated on 0.508 mm Rogers RT/duroid 5880. Fig. 19 shows the photograph and physical dimensions of the fabricated switchable dual-/single-band tunable filter. The dimensions of the lower band channel structures are the same as the C.ABW FA-BPF in Section III. C. Measurement By applying one-bias Vt-lower to lower band varactors Ct a and one-bias Vt-upper to upper band varactors Ct b , two highly selective passband can be individually controlled. Fig. 20(a) and (b) present the measurement and EM simulation results of the demonstrative FA-BPF for the dualband operation. Excellent agreement is obtained for both the lower band and upper band. As shown in Fig. 20(a), by adjusting Vt-lower from 0 to 25 V and keeping Vt-upper at 25 V, the first passband is tuned from 0.76 to 1.78 GHz with C.ABW 79±9 MHz while the second passband is kept fixed at 2.63 GHz. As shown in Fig. 20(b), when Vt-lower is fixed at 0 V and Vt-upper is adjusted from 0 to 25 V, the first passband is kept

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Fig. 21. Measured insertion loss, ABW1 dB , and FBW1 dB versus frequency.

Fig. 22. Measured responses of the dual-band FA-BPF with different power levels of input signal.

unchanged at 0.76 GHz and the second passband is tuned from 1.6 to 2.63 GHz with a fixed FBW1 dB 5.05% ± 0.25%. Four self-adaptive transmission zeros move with two passbands tuning resulting in high selectivity for both two passbands over the entire tuning range. Note that the ABW deviation of the first passband is smaller than the filter in Section III because the stronger source/load coupling and the loading effect further restricts the ABW variation. Fig. 21 summarizes the dual-band operation performance of the demonstrative filter. As observed, the insertion losses of the first band (0.76–1.78 GHz) and the second band (1.6–2.63 GHz) vary from 4.224 to 2.2 dB, and from 4.78 to 2 dB, respectively. The ABW of the first band is 79 ± 9 MHz and the FBW of the second band is 5.05% ± 0.25%. Fig. 22 presents the measured responses with different input power levels. As shown, the filter can handle 13-dBm input signal over the tuning range without a considerable distortion. The higher frequency, the less distortion there is in the passband response. Fig. 23 presents the measured IIP3 of the two passbands with 2 MHz spaced two tones. The IIP3 varies from 11.65 to 19.38 dBm for the first band, and from 15.11 to 26.65 dBm for the second band. Table III presents the comparisons between the presented dual-band filter with the other high-performance dual-band tunable filters. The proposed filter exhibits the widest frequency tuning range for both the first and second band because the proposed stepped-impedance STDRs provide the wide tuning range and broad stopband space. Meanwhile, the advantages, which are inherited from the single band STDR filters such as the high selectivity, C.ABW/FBW, and the simple design/control procedures, also demonstrate the superiority of the proposed filter.

Fig. 23. Measured IIP3 of the dual-band FA-BPF with 2-MHz spaced frequency two tones.

Fig. 24. Measurement and simulation results of the single-band operation with continuous frequency tuning range from 0.76 to 2.63 GHz (five typical responses for demonstration), while the applied voltages are indicated as v t-lower1 /v t-lower2 /v t-upper1 /v t-upper2 .

Applying different voltages to the two varactors in the same STDR will switch OFF the associated passband, as discussed before. Then, the two passbands of the dual-band FA-BPF can work as a single-band tunable filter with very broad tuning range by switching ON and OFF the tunable passbands. Fig. 24 presents the measured and simulated S-parameters of the switchable dual-/single-band FA-BPFs in the singleband operation state. As can be seen, the frequency tuning range covers 0.76–2.63 GHz (110.3%) and the highly selective passband is continuously tuned with around 15-dB switch-OFF isolation. Meanwhile, the in-band performance of the singleband operation is the same as that of the dual-band operation, which confirms that switching OFF one passband does not affect the other passband performance. D. Further Discussion Fig. 25 presents the measured impedance responses of all-ON state, first band-OFF state, and second band-OFF state. It is seen that, in ON state (blue solid line), the two center frequencies are close to the ideal impedance match point (center point). However, when it works in the OFF states, the centerfrequency shifts to the edge of the Smith chart, which implies considerable impedance mismatch [22]. The dual-band impedance response for the switch-ON and -OFF states is the same as single band’s response. It is seen from Fig. 20, there is an overlapping region between two passband tuning ranges, thus it is possible to set two passbands at the same frequency. However, when they are close together, two passbands will undermine each other instead of merging. This is because the loading effect significantly affects the input impedance when two passband fre-

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TABLE III P ERFORMANCE C OMPARISONS OF M EASURED R ESULTS W ITH O THER D UAL -BAND T UNABLE F ILTERS

Fig. 26.

Frequency response with two very close passbands.

reject them by introducing the p-i-n diode-based switchable stubs [41], [42], or move them out of the stopband by adding extra p-i-n diodes at the end of STDRs [43], [44].

Fig. 25. Measured impedance of the ON state, 1 OFF state, and 2 OFF state when v t-lower1 /v t-lower2 /v t-upper1 /v t-upper2 = 0/0/13/13 V for ON state, v t-lower1 /v t-lower2 /v t-upper1 /v t-upper2 = 0/25/13/13 V for 1 OFF state, and v t-lower1 /v t-lower2 /v t-upper1 /v t-upper2 = 0/0/0/25 V for 2 OFF state. (a) S21 responses. (b) Impedance responses in Smith chart.

quencies are close enough. Fig. 26 presents the response when two passbands operate at the same frequency (1.63 GHz). As can be seen, the insertion loss of the first passband deteriorates and the center-frequency shifts toward lower frequency area, while the second passband collapses completely. Comparing to the single band FA-BPF, the single band operation of the filter has poor stopband rejection (Fig. 25), because the OFF-state resonant poles (e.g., 2.6-GHz spur) appear at the stopband and cannot be completely rejected by the impedance mismatch. Nevertheless, it is possible to further

V. C ONCLUSION This paper has proposed a new type of STDR with the controllable frequency tuning range and predefined BW and selectivity characteristics. The control mechanism and adjustment method of the resonator have been investigated. Based on the proposed STDR, three very simple and practical FA-BPFs with predefined ABW1 dB and highly selective responses have been designed, while their selectivity predefine (or stopband predefine) and Q e control ability have also been investigated. The experimental results of the demonstrative C.ABW, D.ABW, and I.ABW filters have validated the analysis and design method. Moreover, a switchable dual-/single-band FA-BPF has been proposed and investigated to realize one C.ABW tunable channel and one constant FBW tunable channel. The experimental results of the demonstrative filter have confirmed the design theory and analysis and achieved the frequency coverage of the entire China 2G/3G/4G cellular wireless/mobile communication spectrum for the first time. R EFERENCES [1] G. M. Rebeiz et al., “Tuning in to RF MEMS,” IEEE Microw. Mag., vol. 10, no. 6, pp. 55–72, Oct. 2009. [2] B. Perlman, J. Laskar, and K. Lim, “Fine-tuning commercial and military radio design,” IEEE Microw. Mag., vol. 9, no. 4, pp. 95–106, Aug. 2008. [3] M. Sherman, A. N. Mody, R. Martinez, C. Rodriguez, and R. Reddy, “IEEE standards supporting cognitive radio and networks, dynamic spectrum access, and coexistence,” IEEE Commun. Mag., vol. 46, no. 7, pp. 72–79, Jul. 2008. [4] M. Iwamura, K. Etemad, M.-H. Fong, R. Nory, and R. Love, “Carrier aggregation framework in 3GPP LTE-advanced [WiMAX/LTE update],” IEEE Commun. Mag., vol. 48, no. 8, pp. 60–67, Aug. 2010. [5] Z. Shen, A. Papasakellariou, J. Montojo, D. Gerstenberger, and F. Xu, “Overview of 3GPP LTE-Advanced carrier aggregation for 4G wireless communications,” IEEE Commun. Mag., vol. 50, no. 2, pp. 122–130, Feb. 2012.

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[6] Y. Rui, P. Cheng, M. Li, Q. T. Zhang, and M. Guizani, “Carrier aggregation for LTE-advanced: Uplink multiple access and transmission enhancement features,” IEEE Wireless Commun., vol. 20, no. 4, pp. 101–108, Sep. 2013. [7] P. W. Wong and I. Hunter, “Electronically tunable filters,” IEEE Microw. Mag., vol. 10, no. 6, pp. 46–54, Oct. 2009. [8] F. Gentili, L. Urbani, G. Bianchi, L. Pelliccia, and R. Sorrentino, “P-I-N-diode-based four-channel switched filter bank with low-power TTL-compatible driver,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3333–3340, Dec. 2014. [9] S. J. Park and G. M. Rebeiz, “Low-loss two-pole tunable filters with three different predefined bandwidth characteristics,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 5, pp. 1137–1148, May 2008. [10] X. Y. Zhang, Q. Xue, C. H. Chan, and B.-J. Hu, “Low-loss frequencyagile bandpass filters with controllable bandwidth and suppressed second harmonic,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 6, pp. 1557–1564, Jun. 2010. [11] Q. Xiang, Q. Feng, X. Huang, and D. Jia, “Electrical tunable microstrip LC bandpass filters with constant bandwidth,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 3, pp. 1124–1130, Mar. 2013. [12] Z. Zhao, J. Chen, L. Yang, and K. Chen, “Three-pole tunable filters with constant bandwidth using mixed combline and split-ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 10, pp. 671–673, Oct. 2014. [13] B.-W. Kim and S.-W. Yun, “Varactor-tuned combline bandpass filter using step-impedance microstrip lines,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 4, pp. 1279–1283, Apr. 2004. [14] A. Anand, J. Small, D. Peroulis, and X. Liu, “Theory and design of octave tunable filters with lumped tuning elements,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4353–4364, Dec. 2013. [15] C.-C. Cheng and G. M. Rebeiz, “High-Q 4–6-GHz suspended stripline RF MEMS tunable filter with bandwidth control,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2469–2476, Oct. 2011. [16] Y.-C. Chiou and G. M. Rebeiz, “A tunable three-pole 1.5–2.2-GHz bandpass filter with bandwidth and transmission zero control,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 11, pp. 2872–2878, Nov. 2011. [17] P.-L. Chi, T. Yang, and T.-Y. Tsai, “A fully tunable two-pole bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 5, pp. 292–294, May 2015. [18] T. Yang and G. M. Rebeiz, “Tunable 1.25–2.1-GHz 4-pole bandpass filter with intrinsic transmission zero tuning,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 5, pp. 1569–1578, May 2015. [19] W. Tang and J.-S. Hong, “Varactor-tuned dual-mode bandpass filters,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 8, pp. 2213–2219, Aug. 2010. [20] J.-R. Mao, W.-W. Choi, K.-W. Tam, W. Q. Che, and Q. Xue, “Tunable bandpass filter design based on external quality factor tuning and multiple mode resonators for wideband applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 7, pp. 2574–2584, Jul. 2013. [21] J. S. Sun, N. Kaneda, Y. Baeyens, T. Itoh, and Y.-K. Chen, “Multilayer planar tunable filter with very wide tuning bandwidth,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 11, pp. 2864–2871, Nov. 2011. [22] Y.-H. Cho and G. M. Rebeiz, “Tunable 4-pole noncontiguous 0.7–2.1-GHz bandpass filters based on dual zero-value couplings,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 5, pp. 1579–1586, May 2015. [23] F. Lin and M. Rais-Zadeh, “Continuously tunable 0.55–1.9-GHz bandpass filter with a constant bandwidth using switchable varactor-tuned resonators,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 792–803, Mar. 2016. [24] X. Huang, L. Zhu, Q. Feng, Q. Xiang, and D. Jia, “Tunable bandpass filter with independently controllable dual passbands,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3200–3208, Sep. 2013. [25] G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 107–116, Jan. 2013. [26] T. Yang and G. M. Rebeiz, “Three-pole 1.3–2.4-GHz diplexer and 1.1–2.45-GHz dual-band filter with common resonator topology and flexible tuning capabilities,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3613–3624, Oct. 2013. [27] Z. H. Chen and Q. X. Chu, “Dual-band reconfigurable bandpass filter with independently controlled passbands and constant absolute bandwidths,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 2, pp. 92–94, Feb. 2016. [28] C.-X. Zhou, C.-M. Zhu, and W. Wu, “Tunable dual-band filter based on stub-capacitor-loaded half-mode substrate integrated waveguide,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 1, pp. 147–155, Jan. 2017.

[29] J. Xu, W. Wu, and G. Wei, “Novel dual-band bandpass filter and reconfigurable filters using lumped-element dual-resonance resonators,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 5, pp. 1496–1507, May 2016. [30] R. Gómez-García and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 4, pp. 1294–1307, Apr. 2015. [31] D. Psychogiou, B. Vaughn, R. Gómez-García, and D. Peroulis, “Reconfigurable multiband bandpass filters in evanescent-mode-cavity-resonator technology,” IEEE Microw. Wireless Compon. Lett., vol. 27, no. 3, pp. 248–250, Mar. 2017. [32] R. Gómez-García, A. C. Guyette, D. Psychogiou, E. J. Naglich, and D. Peroulis, “Quasi-elliptic multi-band filters with center-frequency and bandwidth tunability,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 3, pp. 192–194, Mar. 2016. [33] N. Kumar and Y. K. Singh, “RF-MEMS-based bandpass-to-bandstop switchable single- and dual-band filters with variable FBW and reconfigurable selectivity,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 10, pp. 3824–3837, Oct. 2017. [34] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York, NY, USA: Wiley, 2001. [35] D. Lu, N. S. Barker, and X. Tang, “A simple frequency-agile bandpass filter with predefined bandwidth and stopband using synchronously tuned dual-mode resonator,” IEEE Microw. Wireless Compon. Lett., vol. 27, no. 11, pp. 983–985, Nov. 2017. [36] D. Lu, B. N. Scott, M. Li, and X. Tang, “Synthesis-applied tunable dualmode BPF with highly selective passband and reconfigurable stopband,” IEEE Trans. Microw. Theory Techn., to be published. [37] R. J. Cameron, R. Mansour, and C. M. Kudsia, Microwave Filters for Communication Systems: Fundamentals, Design and Applications. Hoboken, NJ, USA: Wiley, 2007. [38] F.-C. Chen et al., “Design of wide-stopband bandpass filter and diplexer using uniform impedance resonators,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4192–4203, Dec. 2016. [39] M. A. El-Tanani and G. M. Rebeiz, “A two-pole two-zero tunable filter with improved linearity,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 4, pp. 830–839, Apr. 2009. [40] T. Yang and G. M. Rebeiz, “A 1.26–3.3 GHz tunable triplexer with compact size and constant bandwidth,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 10, pp. 786–788, Oct. 2016. [41] S.-C. Weng, K.-W. Hsu, and W.-H. Tu, “Compact and switchable dualband bandpass filter with high selectivity and wide stopband,” Electron. Lett., vol. 49, no. 20, pp. 1275–1277, Sep. 2013. [42] M. L. Chuang and M. T. Wu, “Switchable dual-band filter with common quarter-wavelength resonators,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 62, no. 4, pp. 347–351, Apr. 2015. [43] J. Xu, “Compact switchable bandpass filter and its application to switchable diplexer design,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 1, pp. 13–15, Jan. 2016. [44] C. Zhu, J. Xu, W. Kang, and W. Wu, “Microstrip switchable filtering power divider with three operating modes,” Electron. Lett., vol. 52, no. 25, pp. 2046–2048, Dec. 2016. Di Lu (S’14) was born in Kunming, China, in 1987. He received the B.S. degree from the Electronic Engineering School, Chengdu University of Information and Technology, Chengdu, China, in 2013. He is currently pursuing the Ph.D. degree at the University of Electronic Science and Technology of China (UESTC), Chengdu, China. Since 2015, he has been a Visiting Student with the University of Virginia, Charlottesville, VA, USA. His current research interests include design microwave filters, tunable filters, frequency multipliers, mixers, millimeter-wave circuits, and RF MEMS. Xiaohong Tang (M’08) received the B.S. and Ph.D. degrees in electromagnetism and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China. He is currently a Professor with UESTC. He has authored or co-authored more than 100 journal and conference papers. His current research interests include microwave and millimeter communication and computational electromagnetics. Dr. Tang was a recipient of several national and provincial awards.

LU et al.: SINGLE-BAND AND SWITCHABLE DUAL-/SINGLE-BAND TUNABLE BPFs

N. Scott Barker (S’94–M’99–SM’13) received the B.S.E.E. degree from the University of Virginia, Charlottesville, VA, USA, in 1994, and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Michigan at Ann Arbor, Ann Arbor, MI, USA, in 1996 and 1999, respectively. From 1999 to 2000, he was a Staff Scientist with the Naval Research Laboratory. In 2001, he joined the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, where he is currently a Professor. He recently co-started the company Dominion Micro Probes Inc., Charlottesville, to develop the terahertz frequency wafer probe technology coinvented by his group at the University of Virginia. He has authored or coauthored over 60 publications. His current research interests include applying microelectromechanical systems (MEMS) and micromachining techniques to the development of millimeter-wave and terahertz circuits and components. Prof. Barker has been serving on the MTT-21 Technical Committee on RF-MEMS since 2000 and was the Committee Chair from 2008 to 2011. He has also served for many years on the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and International Microwave Symposium (IMS) Technical Program Review Committee. In 2011, he served on the Steering Committee, IEEE MTT-S IMS, Baltimore, MD, USA. He was the Technical Program Committee Vice-Chair for the 2014 IEEE MTT-S IMS, Tampa, FL, USA. He was an Associate Editor of IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS from 2008 to 2010. He is currently an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECH NIQUES . He was a recipient of the Charles L. Brown Department of Electrical and Computer Engineering New Faculty Teaching Award in 2006 and the Faculty Innovation Award in 2004, the 2003 National Science Foundation CAREER Award, the 2000 IEEE Microwave Prize, and first Second Place in the Student Paper Competition of the IEEE MTT-S IMS.

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Yukang Feng received the M.S. degree in electrical and computer engineering from Northeastern University, Boston, MA, USA, in 2014. He is currently pursuing the Ph.D. degree at the University of Virginia, Charlottesville, VA, USA. Since 2014, he has been a Graduate Research Assistant with the University of Virginia, focusing on millimeter-wave and terahertz electronic. His current research interests include RF MEMS switches, filters, and reconfigurable RF subsystems.

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1

A New Class of K-Band High-Q Frequency-Tunable Circular Cavity Filter Seunggoo Nam, Boyoung Lee, Changsoo Kwak, and Juseop Lee , Senior Member, IEEE

Abstract— A new type of K-band high-Q frequency-tunable waveguide filters is proposed in this paper. The presented filter structure adopts a new technique for tuning the resonant frequency of each resonator. A dielectric plate is inserted in each resonator and rotating it leads to the frequency tuning. Unlike the conventional frequency tuning methodologies for tunable waveguide cavity filters, the new frequency tuning technique alleviates the electrical grounding issue for tuning devices. In addition, we demonstrate a new design method that allows the filter to have an absolute constant bandwidth in the frequency tuning range without using tunable coupling structures. Index Terms— Bandpass filter, frequency tuning, waveguide cavity filter.

I. I NTRODUCTION

R

ECENTLY, the interest in frequency-tunable RF microwave components has been increasing with the rapid growth of frequency-agile communications systems [1], [2]. Hence, various frequency-tunable reconfigurable filter structures have been proposed [3]–[10]. Typical microwave bandpass filters operating below 10 GHz are able to change their center frequencies electrically by virtue of tuning components such as varactors or microelectromechanical systems (MEMS) switches. For example, a structure capable of adjusting the resonant frequency and bandwidth by using T-shaped microstrip line resonators with varactors is introduced in [3]. Similarly, Tsai et al. [4] proposed a microstrip line structure with varactor-loaded loop-shaped resonators which can adjust the frequency and bandwidth. In addition, the work in [5] shows a filter structure with a pair of transmission zeros capable of tuning the center frequency and bandwidth. The papers mentioned above mainly describe filter structures capable of tuning both the center frequency and the bandwidth. Hence, these filters can be tuned to have a constant bandwidth by virtue of tunable coupling structures. However, lossy tuning

Manuscript received May 2, 2017; revised September 7, 2017; accepted October 28, 2017. This work was supported by the Institute for Information and Communications Technology Promotion (IITP) grant funded by the Korean Government (MSIT) (2014-0-00031), Development of Flexible Payload Technologies for Next Satellite Broadcasting and Communications (Corresponding author: Juseop Lee.) S. Nam, B. Lee, and J. Lee are with the Department of Computer and Communications Engineering, Korea University, Seoul, South Korea (e-mail: [email protected]). C. Kwak is with the RF and Satellite Payload Research Team, Electronics and Telecommunications Research Institute, Daejeon 34129, South Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2778075

devices used in the resonators and coupling structures produce loss. Therefore, a lot of efforts have been made to reduce the insertion loss of frequency-tunable filters with a constant absolute bandwidth by employing static coupling structures. Since the static coupling structures have no tuning devices, it leads to a smaller insertion loss when adopted in filter structures. In [6] and [7], planar frequency-tunable filters with a constant bandwidth are reported. Since the loss of the tuning devices such as varactors increases with the frequency in general, it is not preferred to use such devices in designing frequency-tunable filters operating above 10 GHz. Hence, mechanically tunable filters have been studied extensively in pursuing excellent electrical performances such as low loss. In [8] and [9], a K-band fourth-order waveguide bandpass filter structure capable of varying the center frequency and the bandwidth is reported. It uses two sets of tunable resonators: main resonators and coupling resonators. The coupling resonators are used as tunable coupling structures, which allow to adjust all coupling values. Yassini et al. [10] also show a Ka-band fully reconfigurable filter that has been designed based on cascading a low-pass filter and a high-pass filter. By adjusting cutoff frequencies of the low-pass and highpass filters, the center frequency and the bandwidth of the passband can be reconfigured. Yassini et al. [11] present a Ku-band frequency-tunable filter using TE113 dual mode. In this structure, a micrometer-driven drive plate has been used to adjust the resonator size and the center frequency. In [12], a K-band frequency-tunable waveguide filter with a constant absolute bandwidth is reported. A design method for the slots in the filter structure intended to have a constant bandwidth is described. The frequency tuning of the filters in [8]–[12] are based on adjusting the height of each resonator, and this type of frequency tuning method is required to give careful consideration to stable electrical grounding of the moving parts used for changing the height of each resonator since it is challenging to make the moving parts move freely while having an excellent electrical grounding. Other types of frequency-tunable cavity filters are presented in [13]–[16]. Pelliccia et al. [13] and Huang et al. [14] present electrically tunable waveguide filters using MEMS technology that can be tuned discretely and continuously, respectively. However, they have a relatively low Q-factor than a mechanically tunable filter. Vahabisani et al. [15] show a structure for tuning the resonant frequency using liquid metal. This requires a high electrical conductivity of the liquid metal

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and is slower than electrically tunable filters in terms of tuning speed. Prigaud et al. [16] present a Ku-band frequency-tunable filter using dielectric perturbers. It shows a good frequency tuning capability by means of using dielectric pieces that do not require electrical grounding. However, it is challenging to use the mode of the resonator reported in [16] in designing a K-band filter since the resonator size becomes extremely small. This paper presents a K-band frequency-tunable filter design using a higher-order resonant mode. In general, using a higherorder mode in a filter design gives a higher Q-factor, but a number of neighboring modes exist close to the passband of a filter. Hence, this paper also presents effective measures to suppress undesired neighboring modes. This paper is mainly intended to alleviate the grounding issue of the moving parts used for the frequency tuning. We propose a new method to adjust the resonant frequency of a K-band waveguide cavity resonator and show its application to K-band frequency-tunable filters. The new tuning method inserts a dielectric plate into each resonator and rotates it. Since the dielectric plate is used as a tuning device, no electrical grounding is needed, which enables unsophisticated physical implementation. This paper also presents an analytic method for designing a frequency-tunable constant-bandwidth filter that adopts the presented tuning technique. More specifically, we demonstrate an approach to design the static coupling structures such that their coupling values agree with the theoretical desired values for a constant absolute bandwidth over a frequency tuning range. With the aids of the presented tuning technique and design approach for the coupling structures, we have designed and measured K-band frequency-tunable constant-bandwidth waveguide cavity filters.

Fig. 1. Proposed structure and concept of the frequency tunable cavity resonator.

Fig. 2.

II. R ESONATOR C ONCEPT AND D ESIGN The main idea of this paper is to use a dielectric plate in a cylindrical cavity resonator. Rotating the dielectric plate leads to the frequency tuning of the resonator. One of the unique features of the presented frequency tuning method is that the tuning device, the dielectric plate, does not require electrical grounding. Fig. 1 illustrates the structure and concept of the frequency-tunable cavity resonator proposed by this paper. A hole is formed in the lower part of the cavity resonator to insert a shaft for supporting the dielectric plate. From the perspective of the field leakage, the resonance mode should be determined in a way to minimize the loss due to the hole formed at the center. Taking into account the fact that a hole must be placed at bottom center of the cavity, we have analyzed TEnm1 mode whose field components are zero at the center in theory. The field components of TEnm1 mode of the air-filled cylindrical cavity are given by    πz  pnm ρ Hz = H0 Jn cos(nφ) sin a d    πz  ρ βa H0  pnm Hρ =  Jn cos(nφ) cos pnm a d

Bessel functions and their derivatives.

   πz  pnm ρ −βa 2 n H0 sin(nφ) cos J  )2 ρ n ( pnm a d     2 j kηa n H0 pnm ρ πz  Jn Eρ = sin(nφ) sin  2 ( pnm ) ρ a d    πz  ρ j kηa H0  pnm Eφ = Jn cos(nφ) sin  pnm a d

Hφ =

(1)

where ρ, φ, and z are the cylindrical coordinates, and where Jn and Jn denote nth-order Bessel function of the first kind and its derivative, respectively. When n ≥ 2, Jn and Jn are zero with ρ = 0 as shown in Fig. 2. This indicates that the cavity has null field at its center. Hence, we can create a small hole at the center of the cavity having negligible field leakage. As mentioned above, we insert a thin dielectric plate and rotate it to adjust the resonant frequency of the cavity resonator. In this paper, we adopt the mode nomenclature and equations for the air-filled cavity in describing the cavity containing a thin dielectric plate since its field distribution is similar to that of the air-filled cavity. In order to choose the resonant mode for the filter design, we have compared three modes, TE211 , TE311 , and TE411 , in terms of the frequency tuning range and the Q-factor. For fair comparison, the sizes

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Fig. 4. (a) Resonant frequency of the TE311 mode and Q-factor versus the angle of the dielectric plate. (b) E-field distribution of TE311 mode.

Fig. 3. (a) Single-cavity filter test model. (b) Resonant frequency of TE211 mode and Q-factor versus the angle of the dielectric plate. (c) E-field distribution of TE211 mode.

of the cavity and the dielectric plate have been scaled keeping the dimension ratios constant and having 19.55 GHz in the frequency tuning range. First, Fig. 3(a) shows a doubly loaded resonator structure. Fig. 3(b) shows the resonant frequency and Q-factor of TE211 mode and Fig. 3(c) shows the field distributions of a cavity when a dielectric plate rotates. When the dielectric plate exists

where the electric field is strong, the resonance frequency and the Q-factor are low. On the other hand, the resonant frequency and the Q-factor increase as the dielectric plate moves toward the position where the electric field is relatively weak. In other words, the field distribution is determined by the location of excitation and the resonant frequency changes as the dielectric plate rotates since the electric field intensity varies in the azimuth direction. The frequency tuning range is from 19.35 to 19.76 GHz and the Q-factor is 10 460 when the loss tangent of the dielectric plate is assumed to zero. It is shown in Fig. 3(b) that the Q-factor decreases as the loss tangent increases. In [17], a static high-Q cavity filter is proposed using TE221 mode. However, TE221 mode is not used in this paper since it has a number of neighboring modes that cannot be suppressed with ease. Next, Fig. 4 shows the Q-factor and resonant frequency of TE311 mode. As is the case with TE211 mode described above, it can be seen that the resonant frequency and the Q-factor vary depending on the rotation angle of the dielectric plate. Since the resonator size must be set to be larger when a higher-order mode is used, the Q-factor of TE311 mode resonator is larger than that of TE211 mode resonator. However, the frequency tuning range decreases. Finally, Fig. 5 shows the variations of the resonant frequency and corresponding Q-factor when using TE411 mode

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external and internal couplings of our filter for achieving a constant absolute bandwidth. III. C OUPLING D ESIGN In this section, we describe a design method for a secondorder filter utilizing the resonator structure discussed in Section II. It is designed to have a 20-dB equiripple response over the frequency range from 19.4 to 19.7 GHz. In addition, the filter is specified to have a constant absolute 3-dB bandwidth of 150 MHz when tuned over the frequency tuning range. The normalized coupling matrix for the 20-dB equiripple response having 3-dB points at the normalized frequencies of −1 and 1 is given by ⎤ ⎡ 0 M S1 0 0 ⎢ M S1 0 M12 0 ⎥ ⎥ M=⎢ ⎣ 0 M12 0 M L2 ⎦ 0 0 M L2 0 M S1 = M L2 = 0.8007 (2) M12 = 0.7087 where subscripts S and L denote the source and the load, respectively, and the numbers represent the resonators. The relationship between the normalized coupling value and the denormalized coupling value is given by √ kex = fbw · Mex kin = fbw · Min

Fig. 5. (a) Resonant frequency of TE411 mode and Q-factor versus the angle of the dielectric plate. (b) E-field distribution of TE411 mode. TABLE I C OMPARISON OF M ODES

as the dielectric plate rotates. It can be observed that the Q-factor increases and the frequency tuning range decreases as a higher-order mode is in use. Table I summarizes the frequency tuning range and the Q-factor of the modes discussed above. In this paper, we make a compromise between the frequency tuning range and the Q-factor in determining the resonance mode for designing a frequency-tunable filter. Since this paper is to design a tunable filter covering the tuning range from 19.4 to 19.7 GHz, we have chosen TE311 mode in designing our filter. In the following section, we describe the design methods for the

(3)

where fbw is the fractional bandwidth, and where ex and in represent the external and internal couplings, respectively. The desired denormalized coupling values at different frequencies can be obtained from (3). To obtain a constant bandwidth within the target frequency tuning range (19.4–19.7 GHz), the external coupling value should decrease as the frequency increases. Since the change of the coupling value is very small, it must remain almost constant when the frequency varies. Fig. 6(a) shows a traditional resonator and its external coupling structure made of an aperture. A well-known method to tune the resonant frequency of this type of the resonator is adjusting the height of the resonator [8], [10]. As shown in Fig. 6(b), when the height is physically adjusted by using a plunger, the external coupling value generally increases as the frequency increases. In other words, when designing a frequency-tunable filter employing such frequency tuning approach, it is difficult to obtain a coupling value variation suited for obtaining a constant bandwidth from a static coupling structure. Hence, this paper presents a new method for designing a static external coupling structure that can give a desired coupling value variation over a frequency tuning range. Fig. 7(a) shows the presented frequency-tunable resonator structure along with the external coupling slot. By carefully designing the dielectric plate and the coupling slot, the static coupling structure can have a desired coupling value variation. If we use the frequency-tunable resonator structure proposed in this paper, the coupling values for the constant bandwidth can be properly obtained. Fig. 7(b) shows the coupling values of the external

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Fig. 7. (a) Presented frequency-tunable resonator structure. (b) Coupling values of the external coupling structure. Fig. 6. (a) Traditional resonator and its external coupling structure made of an aperture. (b) External coupling coefficient graph.

coupling structure shown in Fig. 7(a) for various values of the dielectric constant of the dielectric plate. It is shown that the slope of the coupling value variation with respect to the frequency decreases as the dielectric constant increases. For example, when the dielectric constant is set to 1.5, the coupling value tends to increase with the frequency. When the dielectric constant increases to 2.2, the coupling value decreases slightly with the frequency. On the other hand, it decreases dramatically with the frequency when the dielectric constant is set to 4.4. Therefore, the dielectric constant has been set to 2.2 in our filter design. Once the dielectric constant has been chosen for acquiring the desired slope of the coupling value variation, the size of the coupling slot must be adjusted to acquire the desired magnitude of the coupling value. In this paper, we have adjusted the size of the slot by varying its height [h in Fig. 7(a)]. Fig. 7(b) shows a larger slot size gives a larger coupling value. In summary, it is required to be able to control independently the magnitude and the slope of the coupling value in the frequency domain. This paper proposes to control them by properly choosing the dielectric constant of the dielectric plate and the size of the coupling slot. The internal coupling structures can also be designed in a similar fashion. Fig. 8(a) and (b) shows the proposed internal coupling structure and its coupling value, respectively. As is

the case with the external coupling, the slope of the internal coupling value variation can be controlled by the permittivity of the dielectric plate, and the overall magnitude of the coupling value can be adjusted by the size of the coupling slot. The height of the internal coupling slot was set to 6.7 mm to meet the required internal coupling values. The structure and dimensions of the second-order bandpass filter based on the above design process are shown in Fig. 9. The thickness of the dielectric plate (Rogers 5880) was set to 3.175 mm. The following section describes the measurement and analysis results of this filter. IV. M EASUREMENT AND A NALYSIS Fig. 10 shows a second-order bandpass filter fabricated by a commercially available milling process. A pair of WR51 waveguide-to-coaxial adapters is connected at both ends. The first measurement result of this filter is shown in Fig. 11. The center frequency is adjustable from 19.38 to 19.8 GHz, with a bandwidth of 149–158 MHz. However, it shows that the unwanted resonant peaks exist in the lower and upper stopbands, and they deteriorate the stopband performance. In order to reduce this deterioration, we have carried out an analysis on these unwanted resonant peaks. The resonance in the lower stopband is TM111 mode and its magnetic field distribution is shown in Fig. 12. This mode can be moved to a lower frequency by perturbing the wall where the electric

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Fig. 10. process.

Second-order bandpass filter fabricated through a typical milling

Fig. 11.

Measured results of the fabricated second-order filter.

Fig. 8. (a) Presented frequency-tunable resonator structure. (b) Coupling values of the internal coupling structure.

Fig. 9. Structure and dimensions of the second-order bandpass filter (a = 12.95 mm, b = 6.48 mm, c = 16 mm, d = 15 mm, e = 3.175 mm, g = 17 mm, h = 5.6 mm, i = 6.7 mm, j = 9 mm, k = 21 mm, and m = 16 mm).

field is strong. The perturbation can be carried out by inserting small screws such that their direction is along with that of the electrical field. This type of perturbation leads to the variation of the capacitance of a resonator, and it has been used in finetuning of various filters [18]–[24].

TM111 mode and TE011 mode of an unperturbed cavity resonate at the same frequency. However, since the dielectric plate is placed in the resonator, in this paper, these two resonant modes do not resonate at the same frequency. TE011 mode resonates at much lower frequencies and is not in the frequency range of interest. Fig. 13 shows the electric field distribution of the resonant peak in the upper stopband, and its mode is degenerate TE311 mode. In order to suppress this degenerate mode, we have used the method presented in [25]. Kuo et al. [25] describe that a better stopband performance can be obtained by placing a higher-order resonance of each resonator at different frequencies. In [25], a coupled line filter has been designed such that the next higher-order mode resonates at different frequencies. In short, staggering the next higher-order mode results in a better rejection in the upper stopband, and this concept can be applied to our filter structure. This paper developed the methodologies for suppressing the neighboring mode mentioned above, and Fig. 14(a) shows the positions of the screws for suppressing the neighboring modes in the second-order filter structure, for demonstration. In resonator 1, a pair of side screws is inserted by 0.7 mm, and the angle of the dielectric plate is 33°. On the other hand,

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Fig. 12. H -field distribution of resonance mode occurring on the left side of the passband (TM111 ).

Fig. 13. stopband.

E-field distribution of degenerated TE311 mode in the upper

resonator 2 has no side screw and the angle of the dielectric plate is 26°. In this case, the mode of the two resonators for forming the passband resonates at the same frequency, but the degenerate mode resonates at different frequencies. Both resonators have the screws on the top and bottom surfaces to suppress TM111 mode. Fig. 14(b) compares the two filter structures: One without the screws and the other having the screws mentioned above. It is shown that the filter with the aforementioned screws has a better stopband performance. The aforementioned techniques for suppressing the undesired resonant peaks have been applied to a fourth-order filter design, and it is discussed in the following section. V. F OURTH -O RDER F ILTER The method for designing a constant-bandwidth filter and the one for suppressing the undesired resonant peaks are also applicable to a high-order filter design. In a typical fourthorder bandpass filter structure, nine tunable structures (four tunable resonators and five tunable coupling structures) are needed to obtain a constant bandwidth. However, by using the frequency-tunable resonator structure proposed in this paper, it is possible to obtain a constant bandwidth although static coupling structures are adopted.

Fig. 14. (a) Top view and perspective view of the second-order filter containing screws for suppressing neighboring modes. (b) Comparison between the filters with and without screws for suppressing neighboring modes.

In this paper, we design a fourth-order filter capable of having a constant bandwdith of 180 MHz over 19.4–19.7 GHz. The normalized coupling matrix for a fourth-order Chebyshev response with 15-dB return loss is given by ⎤ ⎡ 0 0 0 0 0 M S1 ⎢ M S1 0 M12 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 M23 0 0 ⎥ M12 ⎥ M=⎢ ⎢ 0 0 M23 0 M12 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 M12 0 M L2 ⎦ 0 0 0 0 M L2 0 M S1 = M L2 = 0.8403 M12 = 0.6770 M23 = 0.5424.

(4)

Using the design method presented in Section III, it is possible to determine the physical dimensions of the filter, and they are shown in Fig. 15. In addition, screws are placed for the purpose of suppressing the undesired resonant peaks.

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TABLE II C OMPARISON OF T UNABLE F ILTERS

Fig. 15. Structure and dimensions of the fourth-order bandpass filter (a = 12.95 mm, b = 6.48 mm, c = 16 mm, d = 15 mm, e = 3.175 mm, g = 16.8 mm, j = 9 mm, k = 21 mm, m = 16 mm, n = 16 mm, o = 18.2 mm, p = 5.9 mm, q = 6.75 mm, and r = 6.65 mm).

Fig. 16 shows the fabricated fourth-order bandpass filter. The eight screws on the top and bottom are intended to suppress TM111 mode. Resonators 2 and 3 do not have screws on the upper and lower surfaces since the filter can obtain a sufficiently large suppression although the screws are inserted only in resonators 1 and 4. The screws on the side walls of the resonators are for suppressing the degenerated TE311 mode. All screws have been set only at the beginning of filter tuning, and they have not been adjusted at the time of center frequency tuning. Fig. 17 shows the measured and simulated results of the fabricated fourth-order bandpass filter. It can be seen that this filter maintains almost a constant bandwidth (178–184 MHz) while being tuned from 19.36to 19.7 GHz. The maximum bandwidth variation is 6 MHz in the frequency tuning range. The insertion loss of the filter including the coaxial-towaveguide adaptors is less than 1.8 dB. For tuning the filter from 19.36 to 19.7 GHz, resonators 1 and 4 rotate by 30°, while resonators 2 and 3 rotate by 33°. This is because the

Fig. 16.

Fabricated fourth-order bandpass filter.

Fig. 17. Measured and simulated results of the fabricated fourth-order bandpass filter.

loading effect on two resonators is different from that on the other two resonators. Four stepper motors from Motor bank (part number: NK201-01AT) are used for tuning the

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four resonators. The extracted Q-factor from the measurement of the fabricated filter is from 2600 to 3100. There is a discrepancy between the measured Q-factor and the Q-factor from the full-wave simulation, and it can be attributed to the assumption that the loss tangent of the dielectric piece is 0.0009 and it does not vary with the frequency. This discrepancy can be minimized by using a more accurate loss tangent value at the operating frequency of the filter. Table II summarizes the comparison between the previous similar works and this paper for the reader’s convenience. VI. C ONCLUSION In this paper, we have presented a new frequency-tunable waveguide filter structure having rotating dielectric plates. Unlike conventional frequency-tunable filters, the new structure does not necessitate electrical connection between the tuning element and the resonator. Hence, free movement of the tuning devices is allowed. Furthermore, a design approach for acquiring a constant absolute bandwidth by utilizing static coupling structures has been demonstrated. The new frequency tuning method and filter design approach for obtaining a constant bandwidth has been verified by measurement. R EFERENCES [1] M. Yu, B. Yassini, B. Keats, and Y. Wang, “The sound the air makes: High-performance tunable filters based on air-cavity resonators,” IEEE Microw. Mag., vol. 15, no. 5, pp. 83–93, Jul./Aug. 2014. [2] R. R. Mansour, F. Huang, S. Fouladi, W. D. Yan, and M. Nasr, “High-Q tunable filters: Challenges and potential,” IEEE Microw. Mag., vol. 15, no. 5, pp. 70–82, Jul./Aug. 2014. [3] H.-J. Tsai, B.-C. Huang, N.-W. Chen, and S.-K. Jeng, “A reconfigurable bandpass filter based on a varactor-perturbed, T-shaped dualmode resonator,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 5, pp. 297–299, May 2014. [4] H.-J. Tsai, N.-W. Chen, and S.-K. Jeng, “Center frequency and bandwidth controllable microstrip bandpass filter design using loop-shaped dual-mode resonator,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3590–3600, Oct. 2013. [5] X. Luo, S. Sun, and R. B. Staszewski, “Tunable bandpass filter with two adjustable transmission poles and compensable coupling,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 9, pp. 2003–2013, Sep. 2014. [6] C.-W. Tang, C.-T. Tseng, and S.-C. Chang, “Design of the compact tunable filter with modified coupled lines,” IEEE Trans. Compon., Packag., Manuf. Technol., vol. 4, no. 11, pp. 1815–1821, Nov. 2014. [7] F. Lin and M. Rais-Zadeh, “Continuously tunable 0.55–1.9-GHz bandpass filter with a constant bandwidth using switchable varactor-tuned resonators,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 792–803, Mar. 2016. [8] C. Arnold, J. Parlebas, and T. Zwick, “Reconfigurable waveguide filter with variable bandwidth and center frequency,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 8, pp. 1663–1670, Aug. 2014. [9] S. Nam, B. Lee, B. Koh, C. Kwak, and J. Lee, “K-band fully reconfigurable pseudo-elliptic waveguide resonator filter with tunable positive and negative couplings,” IEICE Trans. Commun., vol. 99, no. 10, pp. 2136–2145, Oct. 2016. [10] B. Yassini, M. Yu, and B. Keats, “A Ka-band fully tunable cavity filter,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4002–4012, Dec. 2012. [11] B. Yassini, M. Yu, D. Smith, and S. Kellett, “A Ku-band high-Q tunable filter with stable tuning response,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 2948–2957, Dec. 2009. [12] C. Kwak, M. Uhm, and I.-B. Yom, “Study on slot irises for tunable filters using TE011 mode,” Electron. Lett., vol. 51, no. 3, pp. 266–268, 2015. [13] L. Pelliccia, F. Cacciamani, P. Farinelli, and R. Sorrentino, “High-Q tunable waveguide filters using ohmic RF MEMS switches,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3381–3390, Oct. 2015.

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[14] F. Huang, S. Fouladi, and R. R. Mansour, “High-Q tunable dielectric resonator filters using MEMS technology,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3401–3409, Dec. 2011. [15] N. Vahabisani, S. Khan, and M. Daneshmand, “Microfluidically reconfigurable rectangular waveguide filter using liquid metal posts,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 10, pp. 801–803, Oct. 2016. [16] A. Périgaud et al., “Continuously tuned Ku-band cavity filter based on dielectric perturbers made by ceramic additive manufacturing for space applications,” Proc. IEEE, vol. 105, no. 4, pp. 677–687, Apr. 2017. [17] B. Yassini and M. Yu, “Ka-band dual-mode super Q filters and multiplexers,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3391–3397, Oct. 2015. [18] G. F. Craven and C. K. Mok, “The design of evanescent mode waveguide bandpass filters for a prescribed insertion loss characteristic,” IEEE Trans. Microw. Theory Techn., vol. MTT-19, no. 3, pp. 295–308, Mar. 1971. [19] J. Lee, M. S. Uhm, and I.-B. Yom, “A dual-passband filter of canonical structure for satellite applications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 271–273, Jun. 2004. [20] H. Hu, K. L. Wu, and R. J. Cameron, “Stepped Circular Waveguide Dual-Mode Filters for Broadband Contiguous Multiplexers,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 139–145, Jan. 2013. [21] S. Bastioli, C. Tomassoni, and R. Sorrentino, “A new class of waveguide dual-mode filters using TM and nonresonating modes,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 12, pp. 3909–3917, Dec. 2010. [22] C. K. Mok, “Design of evanescent-mode waveguide diplexers,” IEEE Trans. Microw. Theory Techn., vol. MTT-21, no. 1, pp. 43–48, Jan. 1973. [23] S. Moon, H. H. Sigmarsson, H. Joshi, and W. J. Chappell, “Substrate integrated evanescent-mode cavity filter with a 3.5 to 1 tuning ratio,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 8, pp. 450–452, Aug. 2010. [24] S.-J. Park, I. Reines, C. Patel, and G. M. Rebeiz, “High-Q RF-MEMS 4–6-GHz tunable evanescent-mode cavity filter,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 2, pp. 381–389, Feb. 2010. [25] J.-T. Kuo, S.-P. Chen, and M. Jiang, “Parallel-coupled microstrip filters with over-coupled end stages for suppression of spurious responses,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 10, pp. 440–442, Oct. 2003.

Seunggoo Nam was born in Seoul, South Korea, in 1989. He received the B.E. degree in computer and communication engineering from Korea University, Seoul, in 2015, where he is currently pursuing the Ph.D. degree in radio communications engineering. His current research interests include K-band frequency tunable filters.

Boyoung Lee was born in Seosan, South Korea, in 1991. He received the B.E. degree in radio engineering from Hanbat National University, Daejeon, South Korea, in 2014. He is currently pursuing the Ph.D. degree in radio communications engineering at Korea University, Seoul, South Korea. His current research interests include tunable RF components for radar and satellite systems.

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Changsoo Kwak was born in Seoul, South Korea, in 1971. He received the B.S. degree in mechanical engineering from Yonsei University, Seoul, in 1996, and the M.S. degree in mechanical engineering and Ph.D. degree in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 1998 and 2013, respectively. From 1998 to 2000, he was with Samsung Aerospace Industries, where he was involved in structural analysis and design. Since 2000, he has been with the Electronics and Telecommunications Research Institute, Daejeon, where he has been involved in the development of communications, the oceans, and meteorological satellite, which is the first geostationary satellite developed by South Korea. He was also involved in the development of output multiplexers and electrical analysis of faceted reflectors. His current research interests include microwave filters, multiplexers, mechanically tunable filters, structural/thermal analysis, and optimization.

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Juseop Lee (A’02–M’03–SM’16) received the B.E. and M.E. degrees in radio science and engineering from Korea University, Seoul, South Korea, in 1997 and 1999, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2009. He joined the Electronics and Telecommunications Research Institute, Daejeon, South Korea, in 2001, where he was involved in the design of passive microwave equipment for Ku- and Ka-band communications satellites. In 2005, he joined the University of Michigan, where he was a Research Assistant and a Graduate Student Instructor with the Radiation Laboratory, and was involved in research activities focused on millimeter-wave radars and synthesis techniques for multiple-passband microwave filters. In 2009, he joined Purdue University, West Lafayette, IN, USA, where he was a Post-Doctoral Research Associate, and was involved in the design of adaptable RF systems. In 2012, he joined Korea University, where he is currently an Associate Professor. His current research interests include RF and microwave components, satellite transponders, wireless power transfer, and electromagnetic theories. Prof. Lee was a recipient of the Graduate Fellowship from the Korea Science and Engineering Foundation, Daejeon, and the Rackham Pre-Doctoral Fellowship from the Rackham Graduate School, University of Michigan. He was also a recipient of the IEEE Microwave Theory and Techniques Society Graduate Fellowship. He is currently an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES .

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Tunable SIW Cavity-Based Dual-Mode Diplexers With Various Single-Ended and Balanced Ports Mohamed F. Hagag , Student Member, IEEE, Mohammad Abu Khater , Member, IEEE, Mark D. Hickle , Member, IEEE, and Dimitrios Peroulis, Fellow, IEEE

Abstract— This paper presents, for the first time, tunable dual-mode substrate integrated waveguide (SIW) diplexers with various single-ended (SE) or balanced (BAL) ports. Dual-mode diplexing reduces the required volume by half while signal routing is achieved by proper coupling sections. Furthermore, the SIW resonators result in low loss and wide tunability. This paper experimentally demonstrates three designs with SE–SE, SE–BAL, and BAL–BAL port configurations. The SE–SE, SE–BAL, and BAL–BAL diplexers can be tuned starting from 2.07, 2.2, and 2 GHz, respectively, with a tuning range of 45%, 57.2%, and 63.5%, respectively. The average measured insertion loss is 1.32 dB for the SE–SE, 1.95 dB for the SE–BAL, and 2.15 dB for the BAL–BAL. The average size of the diplexer is 55×55 mm2 . For the proposed SE–BAL and BAL–BAL diplexers, the measured in-band common-mode rejection is better than 40 dB throughout the tuning range. Index Terms— Balanced (BAL) diplexer, diplexer, dual-mode, evanescent-mode cavity, tunable diplexer, tunable resonators.

I. I NTRODUCTION

D

IPLEXERS play a significant role in frequency division duplexing (FDD) systems. There are many measures associated with the performance of diplexers, such as size, isolation, loss, integration, and tunability. As a result, designing a diplexer that meets all such measures is very challenging. Conventional architectures consist of two bandpass filters (BPF) operating at two different frequencies, combined using a distribution network [1]–[3]. The size and performance overhead caused by the distribution network can be eliminated by using dual-mode diplexing [4]–[8]. On the other hand, substrate integrated waveguide (SIW) resonators often present a good compromise between volume and performance due to their compact size and high quality factors [9]–[11]. This technology has also been successfully utilized to implement tunable designs by integrating high-Q tuners, such as electrostatic MEMS or piezoelectric actuators [12]–[17]. Single-ended (SE) RF front ends are easier to implement compared with balanced (BAL) ones. However, the latter is often chosen to resolve common-mode interference and evenorder nonlinearities caused by spectrum crowding and scaling

Manuscript received July 7, 2017; revised September 22, 2017; accepted November 1, 2017. Date of publication December 12, 2017; date of current version March 5, 2018. (Corresponding author: Mohamed F. Hagag.) The authors are with the School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2777978

of electronics [18]–[20]. However, to the best of the authors’ knowledge, no BAL SIW diplexers have been reported in the open literature thus far. Indeed, only a limited number of research efforts integrate the balun functionality into the diplexer. In [21]–[23], static-frequency SE–BAL diplexers were realized using planar resonators. The best achieved insertion loss (IL), amplitude imbalance, phase imbalance, common-mode rejection (CMR), and isolation were 1.5 dB, 1 dB, 2°, 40 dB, and 46 dB, respectively. On the other hand, the only realization of tunable SE–BAL diplexer was done by using planar stepped impedance resonators and varactors as tuners [24]. While each port can be tuned independently, the associated IL was more than 5 dB, and the tuning range was less than 30%. As for BAL–BAL diplexers, the available literature only shows planar static frequency implementations [25]–[28]. For the first time, tunable SIW-based diplexers, with various SE and BAL capabilities, with state-of-the-art performance are introduced. Furthermore, we introduce a unique dualmode diplexing concept that reduces the required volume by half, compared to previous implementations such as [1]. The presented implementations also allow for wideband tuning (around 50%) while maintaining low IL and high isolation. These concepts are experimentally demonstrated by designing SE–SE, SE–BAL, and BAL–BAL diplexers. The main difference between the three diplexers is in the external coupling structure, which provides the SE and BAL ports. The dualmode operation results in a dependent tuning between the two ports. The SE–SE, SE–BAL, and BAL–BAL diplexers can be tuned starting from 2.07, 2.2, and 2 GHz, respectively, for the low band and 2.71, 2.8, and 2.56 GHz, respectively, for the high band, with a tuning range of 45%, 57.2%, and 63.5%, respectively. The average measured IL is 1.32 dB for the SE–SE, 1.95 dB for the SE–BAL, and 2.15 dB for the BAL–BAL. The size of each one of the diplexers is 0.45λg ×0.48λg for the SE–SE, 0.7λg ×0.6λg for the SE–BAL, and 0.5λg × 0.44λg for the BAL–BAL, where λg is the guided wavelength of a 50- microstrip line at the beginning of the tuning range in the low band. However, the diplexing structures for the three diplexers are almost the same size. For the proposed SE–BAL and BAL–BAL diplexers, the in-band CMR is better than 40 dB throughout the tuning range. First, the design details of the dual-mode diplexing are presented including isolation optimization in Section II. Then, the external coupling design, and diplexer implementation

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Fig. 1. Architecture of dual-mode diplexer. The two channels are created by exciting two modes in a common dual-mode resonator. Inset: Cross section of a dual-mode split-post loaded SIW evanescent-mode cavity.

Fig. 2. Magnetic field distribution of the resonators. (a) Mode I. (b) Mode II. (c) Resonator regions classification based on magnetic field distribution of the modes.

and measurements of the SE–SE, SE–BAL, and BAL–BAL diplexers are demonstrated in Sections III–V. II. D UAL -M ODE D IPLEXING C ONCEPT A. Diplexing Architecture The dual-mode diplexing concept is shown in Fig. 1. The incident wave at the source, port 1, excites two modes in a common dual-mode resonator (resonator A), mixed-modes coupling resonator. Mode II is coupled to resonator B, Mode II coupling resonator, (creating high channel on port 2), while the Mode I is coupled to resonator C, Mode I coupling resonator, (creating low channel on port 3). While resonator B can be designed to support only Mode II, it is designed as a dual-mode resonator to match the Q in all the resonators of the mode of interest. Resonator C is not necessarily dualmode resonator because it has to support only Mode I. The resonators used here are a split-post loaded SIW evanescentmode cavities, shown in Fig. 1 [29], [30]. The magnetic field distribution of both modes is shown in Fig. 2(a) and (b). Mode I, shown in Fig. 2(a), can be excited from anywhere in the cavity except the area between the posts [region II in Fig. 2(c)]. On the other hand, Mode II, shown in Fig. 2(b), can be excited only in between the posts or behind them

Fig. 3. (a) Coupling slots in different regions to excite the mode of interest. (b) Resonant frequency of the modes versus coupling aperture angle and length at fixed R1 = 7.4 mm, g = 10 μm, R3 = 2.5 mm, Dp1 = 4.5 mm, d1 = 2 mm, and W1 = W2 = W3 = 2.5 mm. (c) MFR of the modes at different slot widths versus coupling aperture angle and length at fixed R1 = 7.4 mm, g = 10 μm, R3 = 2.5 mm, Dp1 = 4.5 mm, and d1 = 2 mm.

[regions I and II in Fig. 2(c)]. The magnetic field distribution and the resonant frequency of each mode are affected differently by introducing coupling slots in the corresponding regions shown in Fig. 2(c). B. Resonant Frequency Misalignment Introducing a slot in different regions in Fig. 2(c), shown in Fig. 3(a), results in different resonant frequencies [Fig. 3(b)] and different mode frequency ratios (MFRs) [Fig. 3(c)].

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Fig. 4. Resonant frequency of the modes versus capacitance gap size (g) at fixed R1 = 7.4 mm, R3 = 2.5 mm, Dp1 = 4.5 mm, φ1 = 35°, φ1 = 45°, d1 = 2 mm, Ls = 6 mm, and W1 = W2 = W3 = 2.5 mm. An example is shown in the figure. The same resonant frequency (2.5 GHz) of both modes for resonator B and C occurs at a different gap g.

Fig. 5. Structure of the proposed diplexing schematic. (a) Top view. (b) Cross section of resonator B.

The MFR is calculated using MFR =

f Mode II f Mode I

(1)

where f Mode II and f Mode I are the resonant frequencies of Mode II and Mode I, respectively. Having different MFRs for each resonator gives the advantage of detuning the undesired mode in the high or low channels, which mainly provides the required diplexing. Fig. 4 shows the resonant frequency dependence of the two modes in each resonator on the capacitance gap (g). As illustrated in the figure, when the frequencies of Mode II (the desired mode) in resonators A and B are aligned, their undesired Mode I frequencies are misaligned. Likewise, when the Mode I frequencies of resonators A and C are aligned, their Mode II frequencies are misaligned. The frequency misalignment of the undesired modes provides some degree of isolation between the output ports, as does the appropriate choice of external coupling, which will be discussed in the next section. The schematic of the proposed diplexer is shown in Fig. 5. In order to design a dual-mode resonator (resonant frequency of Mode I and the MFR), first, the dual-mode resonator can be designed as a single-mode resonator setting Dp1 = 0. Based on the desired resonant frequency of Mode I, as in [31], we have   1 l = (2) Z o tan 2π fr c 2π fr Cpost where fr is the resonant frequency of Mode I, l is the post height, c is the speed of light in the used substrate, Cpost is the effective capacitance of the post, and Z o is the characteristic impedance of the coaxial line, which is determined by the post and cavity diameters. Then, as shown in Fig. 6, changing variable Dp1 mainly determines the resonant frequency of Mode II

Fig. 6. MFR and resonant frequency of the Mode I versus the distance between the two posts Dp1 and the distance between the post vias d1 of the common resonator, A, at fixed R1 = 8 mm, φ1 = 35°, W1 = 2.5 mm, and g = 10 μm. For the variation of d1 , Dp1 , and R3 are kept fixed at 4.5 and 4 mm, respectively, while d1 and R3 are kept fixed at 2 and 2.5 mm, respectively, for the variation of Dp1 .

and has a slight effect on the resonant frequency of Mode I. At that point, the external coupling is designed, as shown in the following sections. Since the size of the external coupling slot affects the resonant frequency, the dimensions d1 and Dp1 (in Fig. 5) are adjusted to achieve the desired MFR and resonant frequency. Fig. 6 shows how both the MFR and the resonant frequency of Mode I change with Dp1 and d1 , at fixed g. Increasing Dp1 will decrease the MFR with a slight change in the resonant frequency. However, the MFR starts to increase as the post vias become close to the cavity wall-vias.

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Fig. 7. Inter-resonator coupling coefficient K 12 of the upper and lower channels as a function of resonators’ center-to-center distance, L 1 and L 2 , computed at 2.1 GHz, for the lower channel, and 2.7 GHz, for the upper channel. For each channel, the width of the iris is kept fixed, W r1 = 11.3 mm for the lower channel and W r1 = 7.3 mm the upper channel.

C. Inter-Resonator Coupling Fig. 7 shows the effect of varying the resonators’ center to center distance on the intercoupling coefficient of the channels. The inter-resonator coupling is calculated when each mode is aligned in frequency at different resonators using the method introduced in [32]. The inter-resonator coupling also plays a role in suppressing Mode II in the lower channel. This is due to the fact that Mode II is nearly zero in the direction of this coupling [Fig. 2(b)]. Consequently, as shown in Fig. 7, the coupling of Mode II in the lower channel is practically zero. As a result, making resonator C a single-mode or dualmode does not further improve the isolation as will be shown in the following sections. III. SE–SE D IPLEXER I MPLEMENTATION AND M EASUREMENT In order to achieve an SE or a BAL port, the external coupling structure of a diplexer has to be designed accordingly. As a result, the external coupling structure for each one of the proposed diplexers is presented first, followed by the implementation details and measurements. Each channel of the diplexers is synthesized as a secondorder BPF using the coupling matrix method [32]. The employed coupling coefficients are based on a Chebyshev filter response (for 0.2 dB passband ripple). Then, the diplexers are simulated and optimized using ANSYS high-frequency structure simulator (HFSS). A. External Coupling Fig. 8 shows the external coupling coefficient (K e ) factor of the three resonators versus the size of the coupling aperture, at the beginning of the tuning range. K e is calculated, as in [32], using 1 Ke =  (3) ωo ·τ S11 (ωo ) 4

Fig. 8. Simulated external coupling coefficient K e of the three resonators as a function of coupling aperture length and angle, computed at 2.1 GHz, for the lower channel, and 2.7 GHz, for the upper channel. The slot width is kept fixed, W1 = W2 = W3 = 2.5 mm for the three resonators. A 30-mil thick 4350B Roger substrate is employed.

where τ S11 is the group delay of S11 at resonant radian frequency ωo . As shown in the figure, the external coupling of resonator A couples into both modes, while resonators B and C primarily couple into Mode II and Mode I, respectively. This further improves the isolation between the channels. As can be seen in Fig. 8, the leakage in resonator B from Mode I results in a finite amount of coupling. However, Mode I in resonators A and B occurs at differ frequencies, as shown in Fig. 3(a). As a result, the leaked coupling shows no significant effect on isolation. Fig. 8 also shows that, at all ports, the external coupling aperture excites a single shorted transmission line (to maximize the magnetic field coupling), providing an SE operation. B. Implementation and Results An exploded view of the simulated HFSS model of the SE–SE diplexer is shown in Fig. 9. The cavities and signal substrates are 60- and 30-mil thick RO4350B substrates, respectively. The SE–SE diplexer is fabricated using multilayer PCB process, and the final product is shown in Fig. 10. The total size of the diplexer is 0.45λg × 0.48λg (40 × 43 mm) (λg at 2.1 GHz). The bottom of each resonator is covered with a flexible 1-mil thick silver disk attached to a T216-A4NO-273X piezoelectric actuator using silver epoxy. The actuator is used to tune the resonant frequency of the resonators by changing gap (g) between the posts and the silver disk, as shown in Fig. 5(b). The fabricated physical dimensions are shown in Table I. Fig. 11(a) presents the measured and simulated S-parameters of the proposed SE–SE diplexer at the

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Fig. 9. Exploded view of the simulated HFSS model of the SE–SE diplexer.

Fig. 10. Photograph of the proposed SE–SE diplexer. (a) Top view. (b) Bottom view. (c) Side view. The piezoelectric tuner is removed from the common resonator to show the split post. TABLE I D ESIGN PARAMETERS OF THE SE–SE D IPLEXER Fig. 11. (a) Simulated and measured S-parameters of SE–SE diplexer. (b) Passbands magnification of lower and upper channels associated with the subfigures of (a).

beginning, middle, and the end of the tuning range. The lower channel can be tuned from 2.07 to 3 GHz, and the 3-dB fractional bandwidth changes from 4.35% to 5.9% within

the tuning range. The upper channel is tuned (with the lower channel) from 2.71 to 3.9 GHz with a 3-dB fractional bandwidth changing from 2.5% to 3.6%. Fig. 11(b) shows the magnified ILs for the lower and upper channels at the three measured frequencies. Throughout the tuning range, the IL varies between 1.1 to 1.54 dB for the lower channel and from 1.6 to 2 dB for the upper channel. The measured channel-to-channel isolation is better than 36 dB throughout the tuning range.

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Fig. 13. Photograph of the proposed SE–BAL diplexer. (a) Top view. (b) Bottom view. TABLE II D ESIGN PARAMETERS OF THE SE–BAL D IPLEXER

Fig. 12. (a) Schematic of the BAL external coupling structure showing that BAL excitation creates an electric wall in the coupling slot, which maximizes the magnetic field coupling while common excitation creates magnetic wall, which suppresses the coupling of the magnetic field. (b) Simulated external coupling coefficient K e of the three resonators as a function of coupling aperture length and angle, computed at 2.2 GHz, for the lower channel, and 2.8 GHz, for the upper channel. The slot width is kept fixed, W1 = 2 mm, W3 = 1.2 mm, and W S3 = 1.5 mm, for the three resonators. A 16-mil thick 4350B Roger substrate is employed.

IV. SE–BAL D IPLEXER A. External Coupling The external coupling of the common resonator A is similar to that of the SE–SE one. For resonators B (M-II) and C (M-I), the external coupling is a transmission line, with the coupling aperture below its middle point (symmetry plane), as shown in Fig. 12 [33]. An external excitation to the BAL port can be analyzed as a combination of a differential- and common-mode excitation. A differential-mode excitation creates a maximum current at the coupling point due to the electric wall at the symmetry plane. This maximizes the magnetic field coupling to the cavity. A common-mode excitation results in zero current due to the magnetic wall at the symmetry plane. This suppresses the coupling of the common mode into the resonator. These concepts are demonstrated in Fig. 12(a). Resonator C is designed with a single post (single-mode) to increase the magnetic field at the coupling location, providing

sufficient BAL coupling. This also eliminates the second mode in the resonator. Using dual-mode cavity does not provide sufficient BAL coupling because region III (in Fig. 2) would have weaker magnetic field compared to region I, and small slots are used for BAL excitation to decrease CMR. Fig. 12(b) shows the simulated external coupling values for the SE–BAL resonators as a function of the coupling aperture size. K e is calculated using (1). For the SE port, the group delay of S11 is employed while, for the BAL ports, the group delay of the differential-mode reflection coefficient Sdd11 is employed. Sdd11 is obtained by Sdd11 = (S11 − S21 − S12 + S22 )/2.

(4)

The achieved external coupling covers the required values for the presented design with reasonable coupling size. B. Implementation and Results The SE–BAL diplexer is fabricated using a similar process as described in Section III-B. The materials used are a 60- and 16-mil thick RO4350B substrates for the cavities and the signal substrates, respectively. The total size of the diplexer is 0.7λg × 0.6λg (60 mm × 50 mm) (λg at 2.2 GHz). While the diplexing structure is almost the same size compared to the SE–SE diplexer, the external coupling is adding the extra size. A photograph of the implemented diplexer is shown in Fig. 13. The piezoelectric tuners are identical to the ones used in the SE–SE diplexer. The fabricated physical dimensions are shown in Table II. Each channel is measured separately by

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Fig. 14. Simulated and measured S-parameters of SE–BAL diplexer channels. (a) Lower channel is measured and upper channel is terminated with 100- differential load. (b) Upper channel is measured and lower channel is terminated with 100- differential load. (c) Amplitude and phase imbalance within the 10-dB bandwidth associated with the tuning states shown in (a) and (c).

terminating the other channel in a matched load. First, with matching the upper channel (port 2), Fig. 14(a) presents the measured port 3 (lower channel) S-parameters, and the simulated S-parameters of the proposed SE–BAL diplexer at the beginning, middle, and at the end of the tuning range. The lower channel can be tuned from 2.2 to 3.46 GHz, and the 3-dB fractional bandwidth changes from 2.4% to 3% across the tuning range. Second, with matching the lower channel (port 3), Fig. 14(a) and (c) presents the measured port 2 (upper channel) S-parameters, and the simulated S-parameters of the diplexer at the same tuning states. The upper channel is tuned (with the lower channel) from 2.8 to 4.4 GHz with a 3-dB fractional bandwidth changing from 1.6% to 1.8%. Fig. 14(b) shows the amplitude and phase imbalance within the 10-dB bandwidth associated with the tuning states shown in Fig. 14(a) and (c). Fig. 15 presents the maximum amplitude and phase imbalance within the 3-dB bandwidth of the diplexer channels for several tuning states across its tuning range. In all the cases, the amplitude imbalance is less than 0.5 dB, and the phase imbalance is less than 0.7° for both the channels. In all the measured states, the maximum observed CMR is better than 44 dB for both the channels. Throughout the tuning range, the IL varies between 2.2 dB at 2.2 GHz to 1.7 dB at 3.4 GHz for the lower channel, and from 2.1 dB at 2.8 GHz to 2.7 dB at 4.4 GHz for the upper channel. The maximum leakage measured from port 1

Fig. 15. Measured worst case amplitude and phase imbalance within the 3-dB bandwidth of the SE–BAL diplexer for several tuning states across its tuning range.

to the upper channel (S21 ) within the passband of the lower channel is 28 dB, while the maximum observed leakage to the lower channel (S31 ) within the passband of the upper channel is 42 dB. To measure the isolation, four ports are needed to

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TABLE III D ESIGN PARAMETERS OF THE BAL–BAL D IPLEXER

Fig. 16. Simulated external coupling coefficient K e of the three resonators in the BAL–BAL diplexer as a function of coupling aperture length and angle, computed at 2 GHz, for the lower channel, and 2.7 GHz, for the upper channel. The slot width is kept fixed, W s1 = 1.5 mm, W3 = 1.2 mm, and W S3 = 1.5 mm, for the three resonators. A 10-mil-thick 5880 Roger substrate is employed.

Fig. 17. Photograph of the proposed BAL–BAL diplexer. (a) Top view. (b) Bottom view.

measure the two BAL ports, and a fifth port is needed to align the common resonator (port 1). Due to the four-port limitation in the measurement setup, only simulated isolation is shown in Fig. 14. The isolation is expected to be lower than the leakage since the signal path of the isolation (S32 or S23 ) goes through the three resonators, compared to two resonators in the leakage case. This is supported in the simulation results in Fig. 14. As a result, the leakage values are considered as an upper bound for the isolation. V. BAL–BAL D IPLEXER A. External Coupling The external couplings in the BAL–BAL diplexer for resonators B and C are identical to the external couplings in

Fig. 18.

Simulated and measured S-parameters of BAL–BAL diplexer.

the SE–BAL. The common resonator (resonator A) is differentially excited at region I [from Fig. 2(c)] such that it excites both modes. The external coupling values are simulated as a function of the coupling size, and plotted in Fig. 16. The coupling values cover the design requirements with reasonably sized apertures.

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TABLE IV C OMPARISON W ITH SE–SE D IPLEXERS

TABLE V C OMPARISON W ITH SE–BAL D IPLEXERS

TABLE VI C OMPARISON W ITH BAL–BAL D IPLEXERS

B. Implementation and Results The BAL–BAL diplexer is fabricated using multilayer PCB process. The total size of the diplexer is 0.5λg × 0.44λg (60 × 50 mm) (λg at 2 GHz). Also, the diplexing structure does not exceed in size the one proposed in Section III-B. A photograph of the implemented diplexer is shown in Fig. 17. A 60-mil thick RO4350B substrate is employed as a cavity substrate, and a 10-mil thick RO5880 substrate are employed as a signal substrate. The tuning is done using the same aforementioned manner. The fabricated physical dimensions are shown in Table III. As in Section IV-B, each channel is measured separately by matching the other channel. However, here, for the sake of compactness, upper channel and lower channel results are plotted in one figure. Fig. 18 presents the measured and simulated S-parameters of the proposed BAL–BAL diplexer at the beginning, middle, and the end of the tuning range. The lower channel can be tuned from 2 to 3.27 GHz, and

the 3-dB fractional bandwidth changes from 2.3% to 2.8% within the tuning range. The upper channel is tuned (with the lower channel) from 2.56 to 4.23 GHz with a 3-dB fractional bandwidth changing from 1.2% to 1.4%. In all the measured states, the maximum observed CMR is less than 35 dB for both the channels. Throughout the tuning range, the IL varies between 2.5 to 1.8 dB for the lower channel and from 3 to 2.4 dB for the upper channel. The same difficulty in measuring the isolation exists. However, based on the simulated isolation and the leakage between channels, the measured channel-to-channel isolation should be better than 35 dB, throughout the tuning range. VI. M EASUREMENTS OVERVIEW As shown in Figs. 14 and 18, in terms of CMR, the proposed SE–BAL diplexer is better than the proposed BAL–BAL diplexer. There are three reasons for that. First, the used SMA connectors are lower quality in Fig. 17 compared with Fig. 13.

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This was necessary due to the specific layout and is not related to the actual diplexer. Second, in the case of the BAL–BAL diplexer, semirigid extension cables are used to measure the diplexer, due to the tight distance between the ports. This also increases the imbalance. Finally, the cross-coupling between the ports is higher in the BAL–BAL case as can be seen from the measurements. As shown in Fig. 18, CMR between ports 1 and 3 is better than that between ports 1 and 2. A voltage range (−200 to 200 V) has been used, in all implementation, to achieve the frequency tuning. The variations in the range of g is due to fabrication tolerances. Comparisons between other state-of-the-art SE–SE diplexers, SE–BAL diplexers, and BAL–BAL diplexers are shown in Table IV, Table V, and Table VI, respectively. As shown in Table IV, the proposed SE–SE diplexer has the lowest IL with good tuning range and compact size. As shown in Table V, the proposed SE–BAL diplexer has the lowest IL, highest CMR, and tuning range with relatively compact size and reasonable isolation. Regarding the proposed BAL–BAL diplexer, it is the first presented tunable diplexer that supports BAL outputs in all its ports. VII. C ONCLUSION A novel compact tunable low-loss SIW cavity-based SE–SE, SE–BAL, and BAL–BAL diplexers are presented for the first time. Based on the demonstrated design concepts, any combination of BAL and SE ports can be achieved. The only difference between the three presented designs is in the external coupling structures. The measured isolation, CMR, and tuning range is better than 30 dB, 40 dB, and 45%, respectively, for all presented designs. This diplexer design methodology is feasible for SE or BAL multiband FDD systems. R EFERENCES [1] M. A. Khater, Y.-C. Wu, and D. Peroulis, “Tunable cavity-based diplexer with spectrum-aware automatic tuning,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 934–944, Mar. 2017. [2] E. E. Djoumessi and K. Wu, “Electronically tunable diplexer for frequency-agile transceiver front-end,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 1472–1475. [3] K. Saitou and K. Kageyama, “Tunable duplexer having multilayer structure using LTCC,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 3. Jun. 2003, pp. 1763–1766. [4] H. Ezzeddine et al., “Design of a compact dual-band diplexer with dual-mode cavities,” in Proc. 42nd Eur. Microw. Conf., Oct. 2012, pp. 455–458. [5] V. Radoni´c, V. Crnojevi´c-Bengin, A. Baskakova, and I. Vendik, “Multilayer microwave diplexers based on dual-mode resonators for ISM/WiFi bands,” in Proc. Medit. Microw. Symp. (MMS), Dec. 2014, pp. 1–4. [6] T. V. Duong, W. Hong, Z. C. Hao, W. C. Huang, J. X. Zhuang, and V. P. Vo, “A millimeter wave high-isolation diplexer using selectivityimproved dual-mode filters,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 2, pp. 104–106, Feb. 2016. [7] X. Guan, F. Yang, H. Liu, and L. Zhu, “Compact and high-isolation diplexer using dual-mode stub-loaded resonators,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 6, pp. 385–387, Jun. 2014. [8] E. A. Ogbodo, Y. Wu, and Y. Wang, “Microstrip diplexers with dual-mode patch resonant junctions,” in Proc. 46th Eur. Microw. Conf. (EuMC), Oct. 2016, pp. 1155–1158. [9] R. S. Chen, S.-W. Wong, L. Zhu, and Q.-X. Chu, “Wideband bandpass filter using u-slotted substrate integrated waveguide (SIW) cavities,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 1, pp. 1–3, Jan. 2015.

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[33] M. D. Hickle and D. Peroulis, “A widely-tunable substrate-integrated balun filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 274–277. [34] P.-L. Chi and T. Yang, “Three-pole reconfigurable 0.94–1.91-GHz diplexer with bandwidth and transmission zero control,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 1, pp. 96–108, Jan. 2017. [35] T. Yang and G. M. Rebeiz, “A compact 1.9–3.4 GHz diplexer with controllable transmission zeros, improved isolation, and constant fractional bandwidth,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2016, pp. 1–3. [36] C.-H. Ko and G. M. Rebeiz, “A 1.4–2.3-GHz tunable diplexer based on reconfigurable matching networks,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 5, pp. 1595–1602, May 2015. [37] C.-F. Chen, C.-Y. Lin, B.-H. Tseng, and S.-F. Chang, “A compact tunable microstrip diplexer using varactor-tuned dual-mode stub-loaded resonators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2014, pp. 1–3.

Mohamed F. Hagag (S’12) received the B.S. and M.S. degrees in electrical and computer engineering from the Military Technical College, Cairo, Egypt, in 2007 and 2013, respectively. He is currently pursuing the Ph.D. degree at Purdue University, West Lafayette, IN, USA. From 2009 to 2014, he was with Military Technical College, where he was a Researcher and a Teaching Assistant with the Electronic Engineering Department. His research has focused on different metamaterial applications in microwave and millimeter bands especially microwave components, miniaturized multiband antennas, and radar absorbers. He is currently with the School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University. His current research interest includes reconfigurable RF front-end components based on high-Q cavity resonators. Mohammad Abu Khater (S’02–M’16) received the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, USA, in 2015. He was with Younivate (Qualcomm licensee), Intel Labs, and Qualcom, San Diego, CA, USA, where he was involved in various high-speed and low-power circuits and systems. He is currently a Post-Doctoral Researcher with Purdue University. His current research interests include wireless tunable filter control, adaptive RF front ends, MEMS devices monitoring, low-power supplies, and system-level design. Dr. Khater was a recipient of the Fulbright Scholarship in 2007 and the Magoon Award for Excellence in Teaching from the College of Engineering, Purdue University, in 2012.

Mark D. Hickle (S’11–M’17) received the B.S. degree in electrical engineering from the Missouri University of Science and Technology, Rolla, MO, USA, in 2012, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, USA, in 2016. In 2017, he joined BAE Systems, Inc., Merrimack, NH, USA. While at Purdue University, he was a National Defense Science and Engineering Graduate Fellow. Dr. Hickle was a co-recipient of the First Place Awards of the RF-MEMS Tunable Filter Student Design Competitions at both the 2014 and 2015 IEEE MTT-S International Microwave Symposium and a co-recipient of the First Place Award of the 2015 MTT-S Youtube/YouKu Video Competition.

Dimitrios Peroulis (S’99–M’04–SM’15–F’17) received the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2003. He is currently a Professor of electrical and computer engineering, the Deputy Director of the Birck Nanotechnology Center, and the Graduate Admissions Director with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA. He has co-authored over 300 journal and conference papers. He has been a key contributor on developing high-quality reconfigurable filters and filter synthesis techniques based on tunable miniaturized high-Q resonators. He is also leading unique research efforts in high-power multifunctional RF electronics. His current research interests include reconfigurable electronics, cold-plasma RF electronics, and wireless sensors. Dr. Peroulis was a recipient of the National Science Foundation CAREER Award in 2008. In 2012, he was a recipient of the Outstanding Paper Award from the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (Ferroelectrics Section) and the Outstanding Young Engineer Award of the IEEE Microwave Theory and Techniques Society in 2014. He was also a recipient of 10 teaching awards including the 2010 HKN C. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B. Murphy Award, which is Purdue University’s highest undergraduate teaching honor. His students have received numerous Student Paper Awards and other student research-based scholarships. He has been a Purdue University Faculty Scholar.

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Design Methodologies of Compact Orthomode Transducers Based on Mechanism of Polarization Selectivity Ahmed A. Sakr , Walid Dyab, and Ke Wu, Fellow, IEEE

Abstract— An orthomode transducer (OMT) design is presented and implemented for future RF and millimeter-wave applications. The novelty of the proposed OMT stems from a distinct concept of handling dual-polarized signals on the basis of a polarization-selective coupler (PSC). The theory of the PSC is developed and understood through analyzing its constituent waveguides. Such waveguides are hybrid forms of nonradiative dielectric waveguide and substrate-integrated waveguide. An equivalent circuit model is deduced for modeling dispersion characteristics of the PSC. Consequently, design methodologies supported with comprehensive theoretical study and simulation analysis for a PSC-based OMT are developed and examined. The general design of a PSC-based OMT basically depends on a dielectric slab coupler waveguide with a longitudinal periodic PEC polarizer wall in its middle. Practical limitations for this design is discussed and supported with solutions using a proposed top slot coupler instead of the dielectric slab coupler. Closed-form equations are obtained as design recipes for computing structure dimensions and bandwidth based on operating frequency and the material parameters. The proposed structure is prototyped and measured. Measured results show an acceptable agreement with simulation results. Index Terms— Antenna frontends, orthomode transducer (OMT), polarization-selective coupler (PSC), substrate-integrated nonradiative dielectric (SINRD) waveguide, substrate-integrated waveguide (SIW).

I. I NTRODUCTION

W

ITH the current evolutionary growth of mobile systems and wireless communications demand, augmenting the capacity and the efficiency of such systems has become a stipulated research concern [1]. This required capacity enhancement should not be made at the expense of a system bandwidth which is the main resource in a wireless communication system. Fortunately, millimeter-wave (mm-wave) band with its broad bandwidth from 30 to 300 GHz and beyond could be proposed as a candidate for the 5G network to fulfill the required wideband communication services [2], [3]. The importance of the mm-wave band comes in the company of drawbacks such as propagation loss [4], [5] and multipath Manuscript received July 6, 2017; revised September 7, 2017 and October 31, 2017; accepted November 13, 2017. (Corresponding author: Ahmed A. Sakr.) The authors are with the Poly-Grames Research Center, Polytechnique Montréal, Montréal, QC H3T-1J4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2784822

reflections from ground, buildings and other physical objects [6]. In order to achieve the highest possible exploitation of the mm-wave band, its efficiency must be boosted to overcome its drawbacks. One potential solution for that is the deployment of polarization diversity [7]. Polarization diversity has gained much interest in many applications; thanks to its capability in enhancing the efficiency of transmitting and/or receiving circuitries [8]–[10]. An orthomode transducer (OMT) is the essential component for polarization diversity. It is a waveguide device with three physical ports which is capable of separating or combining two spatially orthogonal signals within the same frequency band simultaneously. In this paper, a planar OMT with detailed design recipe is presented and studied. The proposed planar design is suitable for the upcoming generations of mobile systems and wireless communications. There are many techniques to construct an OMT, but most of them are complicated and nonplanar. This is because in those designs, the concept of separating and combining orthogonal signals is basically depending on internal transitions in the geometry of the structure. The Boifot approach [11] is among the common approaches used in the past along with the septum polarizer [12] and the turnstile junction [13]. Different variations of these designs have been reported recently with new manufacturing technologies based on platelet (multilayer) micromachining strategies [14]–[16] and also using superimposition of few aluminum blocks [17]. Those variations, however, did not introduce a new separation concept of the orthogonal polarizations as compared to the mentioned common approaches [11]–[13]. Relatively, new methods for separating the orthogonal polarizations were introduced. In [18], the new method is based on a tilted T-junction between square (or circular) waveguide and rectangular single mode waveguide. Whereas in [19], the method for separation was based on dividing the common square waveguide into four single-ridged triangular waveguides. Planar OMT designs were proposed based on substrateintegrated waveguide (SIW) technologies [20], [21]. Those designs are similar to the proposed design in being planar, but they are different in the polarization separation mechanism as explained in the following. In this paper, an OMT is proposed, studied, and demonstrated based on an idea which is different from the previous design schemes. The realization of this idea basically depends on two types of couplers, namely, the dielectric slab

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coupler [22], [23] and the Riblet-like coupler [24] or the Rosenberg coupler [25]. These couplers are able to totally couple the excited mode at the coupled port if the coupler has an appropriate coupling length. In order to separate vertically polarized and horizontally polarized modes using these couplers, a polarization-selective wall is longitudinally added in the middle of the coupler. This wall allows coupling just for the horizontally polarized mode while guiding the vertically polarized mode. We name this component here, a polarization-selective coupler (PSC). The full theoretical and parametric analysis of the PSC is presented in [26] while the work presented in this paper is more related to the design methodologies, limitations, implementations, and considering the typical substrate anisotropy. The proposed idea about separating or combining the two orthogonal signals using the selective coupling mechanism simplifies the analysis and design as introduced through this paper. This is achieved by designing only one coupler which acts differently for different polarizations. This simplifies the geometry when compared to other OMT structures which depends on two separate couplers oriented on different planes such as the one in [27]. This also relatively reduces the structure size compared to other OMT designs. For example, the dielectric-based design in [20] has a physical size of 20.9 × 20.7 × 30 mm3 when operating around 30 GHz with a bandwidth of 150 MHz, but the proposed dielectric-filled PSC-based OMT has a physical size of 45 × 18 × 3 mm3 when working around 38 GHz with a bandwidth of 5 GHz. For the air-filled designs, the structure in [17] has a physical size of 34×52×68 mm3 , without the antenna, when operating around 32 GHz with a bandwidth of 42% while the proposed air-filled PSC-based OMT including the horn antenna has a physical size of 23 × 42 × 69 mm3 when operating around 32 GHz with a bandwidth of 23%. From these comparisons, the proposed OMT structure shows an interesting reduction in profile. Moreover, the design in [28] showed an interesting compact OMT, but it cannot be integrated within dielectric substrates in the printed circuit board (PCB) designs while the proposed OMT with its reduced and planar profile (without the metal enclosure designed only for measurement) shows a great possibility for integration with PCB applications. However, the proposed OMT has a relatively limited bandwidth governed by the appearance of modes necessary for coupling as shown later. Complete analysis and the design of the previously mentioned couplers, and accordingly OMTs, are considered in this paper based on nonradiative dielectric (NRD) guide [29] and SIW [30] technologies. By means of those technologies, a compact OMT with planar profile is designed based on theoretical analysis and experimental observation. This paper is organized as follows. In Section II, the PSC operation is analyzed through its constituent waveguides. This analysis is enhanced by developing an equivalent circuit model from which the coupling characteristics are obtained. Dispersion and field distribution of the PSC are discussed as well. Complete design procedure, fabrication limitation, and prototype measurements are all presented in Section III. Finally, conclusions and discussions are given in Section IV.

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Fig. 1. Image-NRD waveguide. (a) Physical structure. (b) Field distribution for LSM01 mode. (c) Dispersion curves.

II. P OLARIZATION S ELECTIVE C OUPLER The NRD and SIW waveguide technologies are used together in their image or half-mode forms in order to construct the proposed PSC. For the image-NRD waveguide shown in Fig. 1(a), it consists of a core dielectric region with relative permittivity εd , width a/2 and thickness b. The propagating wave through the dielectric region is bounded at the right-hand side by a vertical PEC wall, bounded at the left-hand side by another dielectric region with lower dielectric constant εeff . This type of waveguide supports two classes of modes, namely, longitudinal section electric (LSE) mode which is vertically polarized and longitudinal section magnetic (LSM) mode which is horizontally polarized. LSM01 is the mode of interest in this waveguide with the field distribution shown in Fig. 1(b). The main difference between image-NRD waveguide and normal NRD waveguide is that the added image PEC wall suppresses all the even-order LSE modes and the odd-order LSM modes. This yields a dispersion diagram, as in Fig. 1(c) with a less number of modes compared to the normal NRD waveguide [29]. Inserting a periodic PEC wall by plating metallic vias on the left-hand side of dielectric region of the image-NRD waveguide yields what we call image-NRD-SIW waveguide, as shown in Fig. 2(a) [31]. The added periodic PEC wall does not affect the LSM modes, while converting the LSE modes to TE modes of the SIW as shown in the field distribution and dispersion curves of Fig. 2(b) and (c), respectively. The TE10 is the mode of interest in this waveguide. The PSC can be constructed by placing the image-NRD and the image-NRD-SIW guides in parallel with an air gap (as an effective medium) in-between as shown in Fig. 3(a) which is the general form of the PSC. This PSC is supposed to be the

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Fig. 3. PSC. (a) Symbol and physical structure. (b) Field distribution when excited with TE10 and LSM01 modes.

Fig. 2. Image-NRD-SIW waveguide. (a) Physical structure. (b) Field distribution for TE10 mode. (c) Dispersion curves.

basic unit in our proposed OMT structure. One can imagine the wave propagation mechanism through this PSC when exciting the structure with two orthogonally polarized modes simultaneously, namely, TESIW 10 and LSM01 modes, as shown in Fig. 3(b). If the image-NRD-SIW guide is excited with a dual-polarized signal, the vertically polarized signal (TESIW 10 ) propagates only through this guide because it is completely confined by the metallic vias without coupling to the adjacent guide. On the contrary, for the horizontally polarized signal (LSM01 ), the periodic conducting wall is transparent for such a horizontal mode, so this mode penetrates evanescently through the air gap (which represents the effective medium in this case) then couples to the adjacent image-NRD guide, and gets fully received there if it has an appropriate coupling length. So the vertically polarized mode TESIW 10 is received at the end of the image-NRD-SIW guide while the horizontally polarized mode LSM01 is received at the end of the image-NRD guide. Therefore, the function of the PSC appears clearly through the field distribution of Fig. 3(b). The vertically polarized TE10 mode is incapable to get coupled from the image-NRDSIW guide to the image-NRD guide, while the horizontally polarized LSM01 mode has the ability to couple from the image-NRD-SIW guide to the image-NRD guide which is the main function of an OMT. In order to facilitate the analysis and design of the PSCbased OMT, an equivalent circuit model is derived as shown in Fig. 4(a). This equivalent circuit model simplifies the understanding of the PSC operation and facilitates the derivation of closed-form design equations for such structures. The modes of interest to be analyzed by this equivalent circuit are the LSMmn modes which are the essential modes for

Fig. 4. (a) PSC equivalent circuit model. (b) Dispersion curves. (c) Power alternation ratio between guide E (excited) and guide C (coupled).

coupling, while the vertically polarized modes are not coupled due to the existence of the periodic wall as stated earlier, and hence they follow the general rules of a rectangular waveguide. The equivalent circuit consists of two transmission lines connected together and terminated with a short circuit load which represents the vertical enclosing PEC walls. The gray transmission line represents the thin air gap and it has characteristic admittance Yeff and length d while the green transmission line represents the dielectric region and it has characteristic admittance Yd and length a/2. The input admittance of this equivalent circuit can be easily written as Yin = Yeff

Yeff + Yd cot(ua/2) coth(vd/2) Yeff coth(vd/2) + Yd cot(ua/2)

(1)

where Yd = j ωεo εr /u

Yeff = j ωεo /v

(2)

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Fig. 5.

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Different views for the general form of a dielectric-filled PSC-based OMT shown in elevation, side view, plan, and isometric cuts.

and u2 + β 2 +

 nπ 2

= k 2 = ω2 μo εr b  nπ 2 −v 2 + β 2 + = k02 = ω2 μ0 ε0 . (3) b Substituting (2) into (1) yields the dispersion relation for the LSMmn modes of the PSC structure. This dispersion relation can be formulated as uεo sin(ua/2) + vεr cos(ua/2) ± (uεo sin(ua/2) − vεr cos(ua/2))e−vd = 0.

(4)

The negative and positive signs give the solution for the even and odd modes, which are achieved at Yin = 0 and Yin → ∞, respectively. From this dispersion relation, the even and odd components of the propagation constant, βe and βo , can be calculated by substituting the solution of (4) into (3) from which the dispersion curves in Fig. 4(b) are obtained. The dispersion curves contain the even and odd components of the LSM01 and LSM21 modes for a PSC filled with dielectric substrate Rogers RT/Duroid 6010 that has a dielectric constant of 10.2, dielectric loss tangent of 0.0023, width a/2 = 2 mm, thickness b = 1.9 mm, and air gap width d = 0.05 mm. The even and odd components of the LSM01 mode are the necessary components for achieving a forward coupling. The appearance of the even component of the LSM21 mode prevents the forward full coupling and limits the bandwidth to about 9 GHz around the operating frequency at 32 GHz. Then, the values of βe and βo of the LSM01 mode are used to obtain the required length for achieving full coupling L c [32] by applying the following equation: π . (5) Lc = βe − βo The power percentage plot in Fig. 4(c) shows the power alternation of the LSM01 mode between the main imageNRD-SIW guide and the coupled image-NRD guide. It can be clearly observed that there is a critical point of operation at 40 mm at which the full coupling is achieved. The design value

of the coupling length is deduced from (5). The implementation of an OMT based on the PSC is explained in Section III where different design possibilities are discussed. III. OMT D ESIGN BASED ON PSC In this paper, the concept of the proposed PSC facilitates the OMT design to be in a systematic procedure. In this section, the design procedure of the planar OMT device is explained. The OMT structure dimensions are deduced according to the required operating frequency and medium parameters. The general case of a PSC-based OMT is studied in Section III-A where some limitations prevent an accurate physical realization of this structure. In Sections III-B and III-C, these limitations are overcome by using dielectric-filled and airfilled fused couplers, respectively. The air-filled structure is integrated with a horn antenna as a validation for the simultaneous separation, or combination, of the two orthogonal polarizations. A. OMT Design Based on Isotropic Dielectric Material Filling With Air Gap In this section, an isotropic dielectric PSC-based OMT with an air gap is presented. Different views of the complete physical structure are presented in Fig. 5. In this case, matching sections to standard waveguides should be added for the purpose of measurements over the frequency range of interest. The effective medium as described in Figs. 1(a) and 2(a) for the image-NRD and image-NRD-SIW guides is considered to be air. Therefore, an air gap should appear between both waveguides as shown in the side view of Fig. 5. The OMT structure is supposed to be excited by a dual-polarized horn antenna. In practice, the antenna may receive an arbitrarily polarized wave. It will be the function of the OMT to decompose the received wave into vertical and horizontal polarizations. Therefore, it is a must to design one input and two independent output sections, specifically suitable for each corresponding

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polarization. Regarding the input section, it is required to support the TE10 and LSM01 modes in the same frequency range. This restricts the cross section of the input guide to be almost a square. For the output sections, it is required to support single mode operation for each polarization at its corresponding output port. Since the band allocated for the 5G cellular network includes the starting edge of the mm-wave band in 28– 38 GHz, we choose this band as a proof of concept to design this OMT at the center frequency of 32 GHz. For the vertically and horizontally oriented output waveguides, two matching sections to WR28 waveguides have been optimized for integration with the Ka-band components. The dimensions are calculated using the procedure explained in detail below. The PSC has a near-square cross section a/2 × b. These dimensions should be selected carefully to guarantee single for the mode operation for each polarization being TESIW 10 vertically polarized mode with cutoff frequency c fcTESIW = √ (6) a εr 10 and LSM01 for the horizontally polarized mode with cutoff frequency   2  2 1 c 1 + . (7) f cLSM01 < √ 2 εr a b From (6) and (7), the dimensions a and b can be evaluated under the following constraint which guarantees that the operating frequency does not permit the propagation of the next odd LSM21 mode: fcLSM01 < fop < f cLSM21 .

(8)

Based on the previous discussions, for this dielectric PSCbased OMT to operate at 32 GHz using a Roger RT/Duroid6010 substrate with dielectric constant 10.2, a/2 and b should be 2 and 1.9 mm, respectively. The gap width between both waveguides, denoted by d, is selected to have the lowest possible value feasible in the fabrication process in order to guarantee the best coupling between the guides before the occurrence of a complete exponential decay of the LSM01 mode in the air gap. This value should be the same as that of the laser beamwidth used for the substrate micromachining which is 0.05 mm in this case. Using the previously obtained parameters, we can apply them to (5) in order to calculate the longitudinal length of the structure which gives 22.5 mm. The final structure after applying the above-mentioned specified parameters is implemented on the HFSS simulation package from which scattering parameters and field distributions are obtained in Fig. 6(a) and (b) for the vertically and horizontally polarized modes, respectively. It can be clearly observed that the vertically polarized TE10 mode is received at port 2 with a complete isolation from the horizontally polarized LSM01 mode. On the other hand, the horizontally polarized LSM01 mode is received at port 3 with an acceptable isolation from the vertically polarized TE10 mode. This attractive feature reflects what is expected through the previous discussions. Nevertheless, some problems will appear if this

Fig. 6. Scattering parameters and field distribution for the structure in Fig. 5 at: (a) port 2 for TE10 mode and (b) port 3 for LSM01 mode.

structure is implemented experimentally and would affect the results, so this should be taken into consideration. The first problem appears because of the laser beam used in Poly-Grames Research Center for cutting the substrate in order to realize the air gap. Since the thickness of the used substrate is relatively large, 1.9 mm, it is really difficult to have a homogeneous air gap over this thickness which results in a rough-tapered air gap. This inhomogeneity causes some reflections and even standing waves may appear as well. The other and main problem is related to realizing the metallic vias. According to the fabrication limitations in many labs, filling the vias with conductor restricts the vias diameter to be at least a half of substrate thickness. This restriction contradicts with our last mentioned restriction for the waveguide cross section to be almost square. In other words, half of the main waveguide cross section will be occupied just for realizing the metallic vias. This affects dramatically the TE10 mode where high reflections take place due to a huge periodic conducting wall. An interesting solution to avoid the previously mentioned restrictions is to nullify the spacing d in Fig. 3(a) altogether

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and make it simply 0. That is, the image-NRD and imageNRD-SIW guides touch each other, and then the resulting coupler is converted from a dielectric slab coupler to a fused coupler instead. The detailed operation, design steps, and physical implementation of this structure are presented in Section III-B. The last issue, which should not be considered as a problem if carefully taken into consideration, is the substrate anisotropy. For the substrates with relatively large dielectric constants, such as the one in this example, there is no escape from the anisotropy effect [33], especially for dual-polarized structures. This is because there are two orthogonal modes propagating through the substrate which means each mode experiences a different dielectric constant which should be taken into account. Considering this, anisotropy extremely complicates the analysis if the air gap exists. Moreover, the dielectric substrates are commonly used for the vertically polarized dominant TE10 mode, so the substrate dielectric constant is usually known in that direction, while in the perpendicular direction, the exact values of the substrate parameters are not specified precisely. For example, regarding the Roger RT/Duroid 6010 substrate used in this paper, Rogers Corporation officially reported to the authors that the dielectric constant in the horizontal direction for this substrate ranges between 13 and 14. This cannot lead to accurate design calculations. In addition, frequency dispersion in connection with dielectric substrate may come into play at mm-wave frequencies, which may yield a different value of dielectric permittivity compared to the commercial datasheet. This requires adding some modification or tuning to the design to facilitate the consideration of such an anisotropy effect. B. OMT Design Based on Gapless Anisotropic Dielectric Material Filling For the discussion mentioned in Section III-A, if the air gap disappears resulting in a homogeneous dielectric substrate, the coupler converges to be a fused coupler. The proposed fused coupler acts as a Riblet junction [24] for the polarization tangential to the slot, while the same coupler acts as a Rosenberg junction [25] for the polarization perpendicular to the common wall. Adding the periodic PEC wall at the center of the coupler preserves the same explained coupling mechanism for the TE01 mode while prevents the coupling of the TE10 mode. Based on that, the OMT functionality is achieved. Thus, this modified version of the proposed coupler, with the periodic PEC wall and the square cross section of input–output waveguides, can be called as polarization-selective top slot coupler, since the slot is on top of the tip of electric field vector of the coupled TE01 mode. In this case, the above-mentioned problems in the design with the air gap can be handled because the PSC unit is simplified to be a single dielectric substrate enclosed in a rectangular waveguide plated with metallic vias longitudinally in its middle as in Fig. 7(c). Since the PSC-based OMT handles two orthogonal signals, the substrate anisotropy [33] should be taken into

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consideration. In this case, the input waveguide cross section is not restricted to be square due to different values of the relative permittivity in the cross section and the longitudinal directions ε X Y and ε Z , respectively. The selected operating frequency f op in this design is 38 GHz and the cross-sectional dimensions can be obtained from the following equations: c c a/2 = 1.2 √ b = 1.2 √ . (9) 2 ε Z f op 2 ε X Y fop The metallic vias can take now an appropriate diameter suitable for the conductor filling process which was not possible in the general case due to the existence of the air gap. Now, the whole PSC width can be considered as a + g where g is effectively considered as half of the vias diameter. This is because when adding the metallic vias with their relatively large diameters according to the plating restriction mentioned earlier, reflections could affect the transmission of the guided TE10 mode. In order to overcome this problem, the width of the common waveguide with length L c should be enlarged with at least a half via diameter which reasonably results in neglecting the caused reflections. The rectangular waveguide filled with an anisotropic material is solved in [34] where the optic axis is aligned with the longitudinal direction of the waveguide. Here, the optic axis of the dielectric substrate is perpendicular to the plane of the substrate, i.e., perpendicular to the direction of propagation. In this case, the dispersion characteristics can be obtained by applying the following relations:   (10) det |k|2 I−kk−ko2 ε¯ r = 0    nπ  mπ aˆ z . (11) k= aˆ x +β y aˆ y + a+g b In our solution using the commercial substrate Rogers RT/Duroid 6010, the permittivity in the z-direction ε Z is different from that of the xy plane ε X Y [35]. Therefore, the optic axis in our design is the y-axis which means that the solution inside the waveguide cannot be written as TE y or TM y as in [36]. Thus the coinciding dispersion curves, TE11 and TM11 which has a cutoff of 36 GHz in this example, will split into two separate modes, as shown in Fig. 7(a). We call them ordinary mode and extraordinary mode with the order“11.” The necessary modes for coupling are the TE01 mode and the extraordinary component of the “11” mode because its main component is horizontally polarized. The next mode which has a major horizontal polarization is the extraordinary mode with order “21,” thus the bandwidth of this structure is between 35 and 41 GHz. Based on this discussion, the coupling length relation in (5) can be rewritten in the following form according to the modes necessary for coupling: π . (12) Lc = β y (TE01 ) − β y (Extraordinary Mode11 ) The variation of the coupling length over the operating bandwidth is shown in Fig. 7(b) where the optimal coupling length at 38 GHz is 9.75 mm. Then, the field distribution and the scattering parameters of the designed structure are shown in Fig. 7(c) and (d), respectively. A complete separation between the vertically polarized TE10 mode and the

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Fig. 7. PSC-based OMT filled with anisotropic dielectric material (Roger RT/Duroid 6010). (a) Dispersion curves. (b) Full coupling length versus frequency. (c) Field distribution. (d) Simulated S-parameters.

horizontally polarized TE01 mode is achieved with an isolation between the two output ports. Fig. 7(d) shows the simulated S-parameters of the OMT where the labels in the graph indicate S (input port number: mode number, output port number: mode number). The port numbering is presented in Fig. 7(c) and mode 1 represents the TE01 mode while mode 2 represents the TE10 mode. It can be observed from the results that the input dual-polarized signal is decomposed into two separated orthogonal signals where the vertically polarized TE10 mode is received at port 2 while the horizontally polarized TE01 is received at port 3. In order to measure, not to realize, this structure, matching sections to the standard air filled WR28 waveguide should be designed. Since we make use of a substrate with high dielectric constant to show the anisotropy effect, the required matching sections are hard to realize. Therefore, we chose to use a dielectric substrate with a lower dielectric constant such as Rogers RT/Duroid 6002 where εXY and ε Z are approximately equal 2.94. This is to validate the concept of the gapless dielectric-filled PSC-based OMT. Although the dielectric used

in this implementation is nearly isotropic, it serves to validate the essence of the idea. The overall structure is prototyped as shown in Fig. 8(a) with the parameters specified in the caption where they can be obtained according to the previously explained procedure. The circular vias presented in Fig. 8(a) is plated with a good conducting material through a chemical deposition process. The implemented structure is measured by the vector network analyzer where the S-parameters are obtained. For the measuring mechanism of this structure, port 1 is excited first with a vertically polarized signal, and then, the output is measured at port 2 while the isolation is measured at port 3. Similarly, port 1 is excited with a horizontally polarized signal and the output is measured at port 3 while the isolation is measured at port 2. The results of the latter case are shown in Fig. 8(b) and compared with the simulation results. It is clear from this comparison that there is an acceptable agreement between the simulation and measured results. This prototype has achieved a bandwidth of about 6 GHz where the design frequency is 38 GHz. This bandwidth is

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WR28 cannot support two modes at a time. In order to analyze the vertical polarization, different matching sections suitable for this polarization should be designed. The horizontally polarized mode is the mode that is selected to be tested in this section because it depends on coupling which mainly governs the OMT design while the vertically polarized mode is received through a normal waveguide without coupling and it has excellent S-parameters in the simulation. According to the previous discussion, for testing both polarizations in a simultaneous manner, a square horn antenna is designed for visualizing the simultaneous performances for both polarizations as explained in Section III-C. For sake of simplicity, and to appreciate the generality of the proposed design methodology, the structure integrated with the horn antenna is implemented as an air-filled structure. This is explained in Section III-C. C. Air-Filled OMT Design In this example, there is no filling material inside the conducting enclosure. Thus, the dielectric coupler converges to an air-filled top slot coupler. This fact significantly simplifies the structure analysis and the design steps as well. The overall structure is integrated with a square horn antenna, as shown in Fig. 9(a). Mathematically, the input admittance obtained in (1) reduces to the following expression when the air gap disappears:

Yeff + Yd cot(ua/2) coth(vd/2) Yin = lim Yeff d→0 Yeff coth(vd/2) + Yd cot(ua/2) (13) = Yd cot(ua/2).

Fig. 8. PSC-based OMT filled with isotropic dielectric material (Roger 6002). (a) Prototype and (b) measured versus simulated S-parameters of the coupled mode TE01 . Coupling length L c = 21 mm, optimized matching length L m = 9.2 mm, and the waveguide cross section of each port is 3 mm × 3 mm.

measured in the Ka-band where 40 GHz is the maximum frequency. Therefore, the real bandwidth may be wider than the mentioned bandwidth limited by the appearance of higher order mode [as in Fig. 7(a)]. However, a slightly high insertion loss, S31 , around 2 dB is observed due to the dielectric loss of the substrate. There is a discrepancy about 1 dB between the simulation and measurement results. This is because the loss in the simulation is based on a very small loss tangent of 0.0012 for the Rogers 6002 substrate. This value is reported in the Rogers datasheet with working conditions up to 10 GHz. The material, however, is used in our design around 38 GHz. We believe this is the cause of that observed discrepancy. Furthermore, the isolation is represented by the value of S23 . The proposed OMT shows an isolation which is better than 20 dB over the whole band. In addition, S21 for the horizontally polarized mode is approximately below −20 dB over the operating band as well. The low values of S23 and S21 guarantee avoiding the reception of unwanted polarization. In Fig. 8(a), the structure feeding-tapered junctions are the standard WR28 waveguide to coaxial transitions. The

More interestingly, closed-form relations can be obtained for the even and odd components of the propagation constant βe and βo , respectively, as in the following equations:  π βe (TE01 ) = ko2 − ( )2 (14) b   π 2  π 2 2 βo (TE11 , TM11 ) = ko − − . (15) b a+g So the full coupling length can be easily written in a closed form as follows: π Lc = . (16) 

π 2

2  π  2 ko2 − b − ko2 − πb − a+g The last expression in (16), obtained from the equivalent circuit model, provides an important design engine for any air-filled (or isotropic dielectric filled such as the prototype in Section III-B) PSC-based OMT by knowing the proper dimensions needed for the required operating frequency range. The coupling length closed form in (16) is valid only whenever the modes propagating through the waveguide are only the necessary modes for achieving coupling which are TE01 mode and TE11 (or the degenerate TM11 ) mode in our case. The appearance of the TE21 mode will prevent the occurrence of the expected full coupling. This limits the bandwidth of the

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Fig. 10. Performance analysis for the structure shown in Fig. 9. (a) Simulated field distribution in the receiving case. The polarization state of the incoming wave is arbitrarily chosen to be circular and (b) measured and simulated S-parameters in the transmitting mode at the two input ports, port 1 and port 2.

Fig. 9. Air-filled PSC-based OMT integrated with horn antenna. (a) Isotropic view of the design. (b) Implemented prototype and (c) measurement setup in antenna anechoic chamber.

structure to be between the TE11 mode and the TE21 mode as in the following relation: ⎛ ⎞    2  2 2  2 c 1 1 2 1 ⎠. BW = ⎝ + − + 2 a+g b a+g b (17) The design parameters based on the specified criteria are a/2 = b = 6 mm and L c = 27 mm at f op = 30 GHz. The resulting BW based on (17) is 6 GHz. For sake of completeness, a 20-mm square horn antenna is integrated to the OMT with length of 20 mm. The final structure is implemented through the HFSS simulation and physically fabricated using CNC machining. The physical structure is easy to fabricate because it is only about the implementation of the conducting enclosure. An important question raised is, how the metallic vias can be implemented although there is no dielectric substrate. This is simply treated by embedding metallic cylinders as protrusions from the metallic enclosure itself as in [37] where each cylinder is divided into two equal parts; each is b/2 in length, where one part is connected to the upper half of the enclosure, while the other part is connected to its lower half, as shown in Fig. 9(b). The structure is measured

in the antenna anechoic chamber as a receiving antenna as in the setup shown in Fig. 9(c). In the simulation, the antenna is used in the reception state as shown in Fig. 10(a). An incoming circularly polarized wave is simulated at 30 GHz propagating at angels ϕi = θi = 90°. From the field distribution in Fig. 10(a), a pure vertically polarized wave is received at port 1, while a pure horizontally polarized wave is received at port 2. This means that the incoming circularly polarized wave is decomposed into its linear orthogonal components where the vertical component is received at port 1 while the horizontal component is completely coupled and received at port 2. The same concept occurs with any incoming polarization (linear, circular or elliptical). For a transmission mode operation of the PSC-based OMT, the measured and simulated S-parameters over a band between 28 and 33 GHz are shown in Fig. 10(b). In Fig. 10(b), the absolute value of S11 represents the return loss of the vertical component at port 1, the absolute value of S22 represents the return loss of the horizontal component at port 2 and the absolute value of S21 represents the isolation between the orthogonal components at port 1 and port 2. It can be observed that the return loss for both polarizations is below 12 dB over a bandwidth of 4 GHz with isolation below 20 dB. The spikes which appear over the S11 and S22 curves are caused by the matching sections to the WR28 waveguide for measurements, which is not related to the OMT functionality. This can be

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Fig. 11.

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Simulation and measured results at (a) port 1: azimuth cut, (b) port 2: azimuth cut, (c) port 1: elevation cut, and (d) port 2: elevation cut.

verified from the S-parameters in Fig. 7(d) before adding the matching sections. To validate the full functionality of the prototype, the antenna radiation pattern is measured in an anechoic chamber for the two polarizations at 30 GHz. The measured gain patterns are shown in Fig. 11. The prototype is setup as the receiving antenna where a standard gain horn antenna is used as the transmitting antenna. First, the transmitting antenna sends a vertically polarized wave and the received signal is measured at port 1 while port 2 is terminated with a matched load. The measured gain in this case is plotted as the V-pol curve marked with “♦” versus azimuth and elevation angles in Fig. 11(a) and (c), respectively. Then, the transmitting antenna is used to send a horizontally polarized wave, and the measured signal at port 1 is plotted as the H-Pol curve marked with “O” in Fig. 11(a) and (c) for the azimuth and elevation cuts, respectively. The whole procedure is then repeated while port 1 is terminated with a matched load and the signal is measured at port 2. The measured results in this case are shown in Fig. 11(b) and (d) for azimuth and elevation cuts, respectively. From Fig. 11(b) and (d), it is deduced that the

OMT prototype is functional for all the azimuth and elevation angles with an isolation better than 25 dB at 30 GHz. IV. C ONCLUSION A practical design procedure for the proposed planar OMT is presented and evaluated. The core idea of the proposed OMT depends on the design of a PSC. The PSC consists of two parallel image-NRD and image-NRD-SIW waveguides and studied through a simple equivalent circuit model by which the dispersion characteristics are obtained. The general form of a PSC-based OMT structure with the matching sections is designed, and the fabrication limitations are discussed, namely, the hard-to-realize gap width and the metallic via conductor plating. As a solution for the fabrication limitations, the imageNRD and image-NRD-SIW waveguides are fused into one waveguide. The anisotropy of typical materials is taken into account with an exact analytical procedure. A complete design recipe is explained for obtaining structure dimensions and operation bandwidth for a specific frequency range. This analytical procedure that takes the anisotropy into account is validated through full wave simulations. However, it was not

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physically realized due to the complicated required matching sections to air-filled waveguide transitions. Two prototypes are implemented and measured. The first prototype is to validate the design procedure and concept of the isotropic dielectricbased fused PSC. In this prototype, the air gap is totally eliminated to overcome the hard realization of a uniform gap width. Finally, the dielectric material is removed altogether, getting rid of the dielectric loss at the expense of a larger physical size. This might be suitable to the applications where the sensitivity and reliability are more important than the size. As another proof of concept, this air-filled design is realized and integrated with a square horn antenna. An excellent agreement is achieved with the simulation results. The design of transitions to other transmission lines such as CPW and microstrip lines is considered as a future work. This will facilitate the full integration of the proposed OMT with lowprofile handheld equipment. ACKNOWLEDGMENT The authors would like to thank the reviewers for their efforts to enhance the readability and clarity of this paper’s manuscript. They would also like to thank the technicians of the Poly-Grames Research Center for their help with the fabrication and measurements of the prototypes. R EFERENCES [1] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. [2] M. Elkashlan, T. Q. Duong, and H.-H. Chen, “Millimeter-wave communications for 5G: Fundamentals: Part I,” IEEE Commun. Mag., vol. 52, no. 9, pp. 52–54, Sep. 2014. [3] M. Elkashlan, T. Q. Duong, and H.-H. Chen, “Millimeter-wave communications for 5G—Part 2: Applications,” IEEE Commun. Mag., vol. 53, no. 1, pp. 166–167, Jan. 2015. [4] A. I. Sulyman, A. Alwarafy, G. R. MacCartney, T. S. Rappaport, and A. Alsanie, “Directional radio propagation path loss models for millimeter-wave wireless networks in the 28-, 60-, and 73-GHz bands,” IEEE Trans. Wireless Commun., vol. 15, no. 10, pp. 6939–6947, Oct. 2016. [5] A. I. Sulyman, A. T. Nassar, M. K. Samimi, G. R. MacCartney, Jr., T. S. Rappaport, and A. Alsanie, “Radio propagation path loss models for 5G cellular networks in the 28 GHz and 38 GHz millimeter-wave bands,” IEEE Commun. Mag., vol. 52, no. 9, pp. 78–86, Sep. 2014. [6] H. Zhao et al., “28 GHz millimeter wave cellular communication measurements for reflection and penetration loss in and around buildings in New York city,” in Proc. IEEE Int. Conf. Commun., Jun. 2013, pp. 5163–5167. [7] K. Fujimoto, Mobile Antenna Systems Handbook. Norwood, MA, USA: Artech House, 2008. [8] J. Helander, K. Zhao, Z. Ying, and D. Sjöberg, “Performance analysis of millimeter-wave phased array antennas in cellular handsets,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 504–507, 2015. [9] J. Song, J. Choi, S. G. Larew, D. J. Love, T. A. Thomas, and A. A. Ghosh, “Adaptive millimeter wave beam alignment for dualpolarized MIMO systems,” IEEE Trans. Wireless Commun., vol. 14, no. 11, pp. 6283–6296, Nov. 2015. [10] S. Liao et al., “Passive millimeter-wave dual-polarization imagers,” IEEE Trans. Instrum. Meas., vol. 61, no. 7, pp. 2042–2050, Jul. 2012. [11] A. M. Boifot, E. Lier, and T. Schaug-Pettersen, “Simple and broadband orthomode transducer (antenna feed),” Proc. Inst. Elect. Eng.—Microw., Antennas Propag., vol. 137, no. 6, pt. H, pp. 396–400, Dec. 1990. [12] D. Davis, O. Digiondomenico, and J. Kempic, “A new type of circularly polarized antenna element,” in Proc. Antennas Propag. Soc. Int. Symp., Oct. 1967, pp. 26–33. [13] M. A. Meyer and H. B. Goldberg, “Applications of the turnstile junction,” IRE Trans. Microw. Theory Techn., vol. MTT-3, no. 6, pp. 40–45, Dec. 1955.

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[14] C. A. Leal-Sevillano, K. B. Cooper, J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J. M. Rebollar, “A 225 GHz circular polarization waveguide duplexer based on a septum orthomode transducer polarizer,” IEEE Trans. THz Sci. Technol., vol. 3, no. 5, pp. 574–583, Sep. 2013. [15] C. A. Leal-Sevillano, Y. Tian, M. J. Lancaster, J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J. M. Rebollar, “A micromachined dual-band orthomode transducer,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 1, pp. 55–63, Jan. 2014. [16] G. Virone, O. A. Peverini, M. Lumia, G. Addamo, and R. Tascone, “Platelet orthomode transducer for Q-band correlation polarimeter clusters,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 7, pp. 1487–1494, Jul. 2014. [17] D. Dousset, S. Claude, and K. Wu, “A compact high-performance orthomode transducer for the atacama large millimeter array (ALMA) band 1 (31–45 GHz),” IEEE Access, vol. 1, pp. 480–487, 2013. [18] J. Esteban and C. Camacho-Peñalosa, “Compact orthomode transducer polarizer based on a tilted-waveguide T-junction,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3208–3217, Oct. 2015. [19] J.-H. Hwang and Y. Oh, “Compact orthomode transducer using singleridged triangular waveguides,” IEEE Microw. Compon. Lett., vol. 21, no. 8, pp. 412–414, Aug. 2011. [20] M. K. Mandal, K. Wu, and D. Deslandes, “A compact planar orthomode transducer,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [21] M. Esquius-Morote, M. Mattes, and J. R. Mosig, “Orthomode transducer and dual-polarized horn antenna in substrate integrated technology,” IEEE Trans. Antennas Propag., vol. 62, no. 10, pp. 4935–4944, Oct. 2014. [22] C. Yeh, F. Manshadi, K. F. Casey, and A. Johnston, “Accuracy of directional coupler theory in fiber or integrated optics applications,” J. Opt. Soc. Amer., vol. 68, no. 8, pp. 1079–1083, 1978. [23] A. A. Sakr, W. Dyab, and K. Wu, “Image theory based miniaturization of nonradiative dielectric coupler for millimeter wave integrated circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 463–465. [24] H. J. Riblet, “The short-slot hybrid junction,” Proc. IRE, vol. 40, no. 2, pp. 180–184, Feb. 1952. [25] U. Rosenberg and W. Speldrich, “A new waveguide directional coupler/hybrid type-favorably suited for millimeter wave application,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 3. Jun. 2000, pp. 1311–1314. [26] A. A. Sakr, W. M. Dyab, and K. Wu, “Theory of polarization selective coupling and its application to design of planar orthomode transducers,” IEEE Trans. Antennas Propag., to be published. [27] O. A. Peverini, R. Tascone, G. Virone, A. Olivieri, and R. Orta, “Orthomode transducer for millimeter-wave correlation receivers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 5, pp. 2042–2049, May 2006. [28] U. Rosenberg and R. Beyer, “Compact T-junction orthomode transducer facilitates easy integration and low cost production,” in Proc. 41st Eur. Microw. Conf., Oct. 2011, pp. 663–666. [29] T. Yoneyama and S. Nishida, “Nonradiative dielectric waveguide for millimeter-wave integrated circuits,” IEEE Trans. Microw. Theory Techn., vol. MTT-29, no. 11, pp. 1188–1192, Nov. 1981. [30] D. Deslandes and K. Wu, “Accurate modeling, wave mechanisms, and design considerations of a substrate integrated waveguide,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 6, pp. 2516–2526, Jun. 2006. [31] W. Dyab, A. A. Sakr, and K. Wu, “Characterization of substrate integrated non radiative dielectric slab waveguide for cross-polarized mmwave components,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 1798–1800. [32] R. Mongia, I. J. Bahl, P. Bhartia, and S. J. Hong, RF and Microwave Coupled-line Circuits. Norwood, MA, USA: Artech House, 2007. [33] J. Coonrod, “General information of dielectric constants for circuit design using Rogers high frequency materials,” Rogers Corporation, Rogers, CT, USA, Tech. Rep., 2010. [34] S. Liu, L. W. Li, M. S. Leong, and T. S. Yeo, “Rectangular conducting waveguide filled with uniaxial anisotropic media: A modal analysis and dyadic Green’s function,” J. Electromagn. Waves Appl., vol. 14, no. 1, pp. 45–47, Jan. 2000. [35] J. C. Rautio and S. Arvas, “Measurement of planar substrate uniaxial anisotropy,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 10, pp. 2456–2463, Oct. 2009. [36] J. A. Kong, Electromagnetic Wave Theory. Hoboken, NJ, USA: Wiley, 1990. [37] A. Navarrini and R. Nesti, “Symmetric reverse-coupling waveguide orthomode transducer for the 3-mm band,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 1, pp. 80–88, Jan. 2009.

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Ahmed A. Sakr received the B.Sc. degree (with distinction) in electronics and electrical communication and M.Sc. degree in engineering physics from the Faculty of Engineering, Cairo University, Giza, Egypt, in 2010 and 2014, respectively. He is currently pursuing the Ph.D. degree at the Department of Electrical Engineering, Polytechnique Montréal (University of Montréal), Montréal, QC, Canada. From 2011 to 2014, he was a Teaching and Research Assistant with the Department of Engineering Physics, Cairo University. He is a currently a member of the Poly-Grames Research Laboratory, Polytechnique Montréal. His current research interests include the design of dual-polarized millimeterwave integrated circuits, antennas, computational electromagnetics, slow-wave structures, and electromagnetic modelling of composite materials. Mr. Sakr was a recipient of the PERSWADE Scholarship Award funded by the Natural Sciences and Engineering Research Council of Canada for his Ph.D. studies.

Walid Dyab was born in Alexandria, Egypt, in 1981. He received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Alexandria, Alexandria, in 2003 and 2007, respectively, and the Ph.D. degree in electrical and computer engineering from Syracuse University, Syracuse, NY, USA, in 2014. From 2003 to 2005, he was a Teaching Assistant with the Alexandria Institute of Engineering and Technology, Alexandria. From 2005 to 2006, he was a Technical Support Engineer with the Radio Network Department, Alcatel, Egypt. From 2006 to 2009, he was a Teaching and a Research Assistant with the German University, Cairo, Cairo, Egypt. From 2012 to 2014, he was a Teaching and a Research Assistant with the Department of Electrical Engineering, Syracuse University. Since 2015, he has been a Post-Doctoral Fellow with the Ecole Polytechnique de Montréal, Montréal, QC, Canada. His current research interests include electromagnetic theory, antennas and electromagnetic wave propagation, propagation of electromagnetic waves on top of imperfect ground planes, antenna measurements, adaptive antenna systems, adaptive signal processing, design and analysis of microwave passive circuits, time reversal electromagnetics, and electromagnetic surface waves. Dr. Dyab was the recipient of the Doctoral Research Award of the IEEE Antennas and Propagation Society in 2012 and the FRQNT Postdoctoral Fellowship Award from the Government of Quebec in 2016. He held the Syracuse University Graduate Fellowship Award from 2009 to 2012.

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Ke Wu (M’87–SM’92–F’01) received the B.Sc. degree (with distinction) in radio engineering from Southeast University, Nanjing, China, in 1982, and the D.E.A. and Ph.D. degrees (with distinction) in optics, optoelectronics, and microwave engineering from the Institut National Polytechnique de Grenoble, University of Grenoble, Grenoble, France, in 1984 and 1987, respectively. He was the Director of the Poly-Grames Research Center, Montréal, QC, Canada. He was the Founding Director of the Center for Radiofrequency Electronics Research of Quebec (Regroupement stratégique of FRQNT) and a Tier-I Canada Research Chair of RF and Millimeter-Wave Engineering. He has held guest, visiting, and honorary professorships with many universities. He is currently a Professor of electrical engineering and an NSERC-Huawei Industrial Research Chair of Future Wireless Technologies with the Polytechnique Montréal (University of Montreal), Montréal. He is also with the School of Information Science and Engineering, Ningbo University, Ningbo, China, on leave from his home institution, leading a special 5G and future wireless research program. He has authored or co-authored over 1100 referred papers and a number of books/book chapters. He has filed more than 50 patents. His current research interests include substrate-integrated circuits, antenna arrays, field theory and joint field/circuit modeling, ultrafast interconnects, wireless power transmission and harvesting, megahertz-through-terahertz transceivers and sensors for wireless systems and biomedical applications, and modeling and design of microwave and terahertz photonic circuits and systems. Dr. Wu is a Fellow of the Canadian Academy of Engineering and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of the Electromagnetics Academy, Sigma Xi, URSI, and IEEE Eta Kappa Nu. He was a recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award, the 2004 Fessenden Medal of the IEEE Canada, the 2009 Thomas W. Eadie Medal of the Royal Society of Canada, the Queen Elizabeth II Diamond Jubilee Medal in 2013, the 2013 FCCP Education Foundation Award of Merit, the 2014 IEEE MTT-S Microwave Application Award, the 2014 MarieVictorin Prize (Prix du Quebec—the highest distinction of Québec in the natural sciences and engineering), the 2015 Prix d’Excellence en Recherche et Innovation of Polytechnique Montréal, and the 2015 IEEE Montreal Section Gold Medal of Achievement. He has held key positions in and has served on various panels and international committees including the Chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. In particular, he was the General Chair of the 2012 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium. He served on the Editorial/Review Boards of many technical journals, transactions, proceedings, and letters, as well as scientific encyclopedia, including as an Editor or a Guest Editor. He was the Chair of the joint IEEE chapters of MTT-S/AP-S/LEOS, Montreal, QC, Canada. He is currently the Chair of the newly restructured IEEE MTT-S Montreal Chapter. He was an elected IEEE MTT-S Administrative Committee (AdCom) member from 2006 to 2015 and served as the Chair of the IEEE MTT-S Transnational Committee, the Member and Geographic Activities Committee, and the Technical Coordinating Committee among many other AdCom functions. He was an IEEE MTT-S Distinguished Microwave Lecturer from 2009 to 2011. He was the 2016 IEEE MTT-S President. He is the Inaugural Representative of North America as a member of the European Microwave Association General Assembly.

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1

Compact Design of Planar Quadrature Coupler With Improved Phase Responses and Wide Tunable Coupling Ratios Feng Lin , Member, IEEE

Abstract— This paper presents a compact planar tunable quadrature coupler with improved phase responses. Wide coupling-tuning ratio is achieved by using two varactors loaded on the center of the transmission lines of the modified coupler. Closed-form equations are derived for design parameters. For verification, a 1-GHz tunable coupler is designed and measured. The measured results agree well with the simulated ones. The measured power-dividing ratio can be tuned in a range from 16.2 to −35 dB (from 42 to 3.2E-4) with better than 20-dB return loss and isolation, while the phase imbalance is smaller than 10°. The coupler size is 0.21λg × 0.08λg and reduced by 73.1% compared with the conventional branch-line coupler. The theoretical analysis shows that the phase imbalance and insertion loss are mostly caused by the loss from varactors. Then, a tunable coupler with improved phase and loss responses is proposed, where the additional phase difference and insertion loss resulting from the varactors are compensated for by introducing a negative resistance from the negative impedance converter. Measured results of a demonstrative 1-GHz coupler show a power-dividing ratio tuning range from 24.4 to −22.2 dB (from 275.4 to 6E-4) while maintaining 20-dB return loss and isolation. The phase imbalance is smaller than 1° and the insertion loss is improved by 1 dB and nearly close to theoretical values across the tuning range. Index Terms— Coupler, microstrip, planar circuits, powerdividing ratio, reconfigurable.

I. I NTRODUCTION

Q

UADRATURE couplers [1], which exhibit two 90° phase-difference outputs with perfect isolation and all ports matched, are frequently used in microwave and millimeter-wave systems. In the past, quadrature couplers were extensively studied for multiband/broadband [2], [3], size miniaturization [4], and arbitrary power-dividing ratio [5] applications. There is an increasingly high demand for compact, low-loss reconfigurable couplers in various advanced wireless applications: adaptive antennas, multiport amplifiers [6], and modern reconfigurable transceivers. The Manuscript received July 9, 2017; revised September 25, 2017 and October 27, 2017; accepted November 13, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61601026 and in part by the Open Project of the State Key Laboratory of Millimeter Waves under Grant K201701. The author is with the School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China, and also with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2783375

quadrature couplers with tunable coupling ratio can find applications in the design of polarization [7] and pattern [8] reconfigurable antennas. Several designs have been reported for implementing reconfigurable couplers. Planar transmission-line couplers [9]–[16] offer wider tuning range and a simple structure compared to other alternative implementations, such as lumped LC couplers [17]–[19] and microelectromechanical systems (MEMS) couplers [20]. For lumped LC couplers, the main tuning elements are achieved by CMOS active inductors and capacitors [17]–[19]. A CMOS coupler tuning from 2 to 6 GHz with coupling coefficients from 1.3 to 9 dB was proposed in [18] by using varactors and tunable active inductors. By mechanically changing the geometry of 3-D micromachined coupled transmission lines with integrated MEMS actuators, two coupling states of the MEMS coupler with bandwidth from 10 to 18 GHz were realized in [20]. Similar to the design of tunable power divider [11], there are also three basic design methods for planar tunable transmission-line couplers. The first one is the switched transmission-line configuration, which uses switch networks to select different feeding paths to vary the power-dividing ratio [12], [13]. In [13], the power-dividing ratio of the quasi-lumped coupler can be tuned among the four operation modes by controlling the p-i-n diodes. The drawback of this method is the limited tuning states and complicated design for antenna array applications. The second method is varying the transmission-line characteristic impedance of the couplers to tune the coupling ratio. In [14], a 3.5-GHz patch hybrid coupler was demonstrated with a continuously tuned coupling from −4.2 to −10 dB by varactors loaded transmission lines. However, for this method, the coupling tuning range is limited by the tuning range of characteristic impedances to ensure the port isolation and matching. In the third type, the couplers are designed as special circuit structures by using varactor diodes to simultaneously realize continuous coupling tuning and good matching and isolation [15], [16]. In [16], a two-section branch-line coupler with a coupling from 0 to 20 dB was proposed by tuning varactors. The port matching and isolation are improved while obtaining a wide tuning range. On the flip side, the circuit needs to be further miniaturized, and its tuning range is mainly limited by the phase imbalance ( 0, ω2 L e C gd < 1.

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Fig. 13.

7

Simulated S-parameters versus different capacitances C D . (a) |S11 |. (b) |S21 |. (c) |S31 |. (d) |S41 |. (e) Phase differences.

From (24), the frequency range showing the negative resistance can be obtained as 2π

1 < f < √ L d Cd

 2π L d

1 

C d C gd C d +C gd

.

(25)

When designing the feedback circuit, L d and Cd are first determined to make sure the designed frequency is in the frequency range showing negative resistance. Then, one can adjust Rd in the consideration of stability and negative resistance value. Fig. 10(c) shows the layout configuration of the improved tunable coupler implemented on a single substrate. The dc blocking capacitors Cblock (100 pF), fixed capacitors C0 (3.6 pF), and C1 (4.7 pF) are realized by ATC 600S series capacitors. The varactors C D and transistors are implemented by MA46H202 GaAs diodes (C D , from 0.7 to 10 pF, Rs = 1  in simulation) and Avago Technologies ATF53189 transistors, respectively. The negative resistance is obtained at 1 GHz by choosing Cd of 6 pF, L d of 7 nH, and Rd of 2  in the feedback structure. The bias circuit for ATF53189 is realized using two 47-nH inductors (L D and L G ). The dc blocking capacitor C g of 100 pF is connected in series with varactor to isolate the bias voltage from the varactor. Inductor L g is set to 2.5 nH to guarantee that the capacitance tuning

Fig. 14. Comparison of the simulated results versus power dividing ratio at f 0 of the proposed tunable coupler with and without negative impedance converter. (a) −90° mode. (b) 90° mode.

range of varactor C D will not be interfered with. The bias circuit for C D is realized using one 180-k resistor (Rbias). Fig. 11 shows the simulated resistance values of the negative

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Fig. 15.

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Measured S-parameters for different power dividing ratios. (a) |S11 |. (b) |S21 |. (c) |S31 |. (d) |S41 |. (e) Phase differences.

impedance converter and stability factors of the whole coupler for different capacitances C D . The simulated real part of input impedance [Z in in Fig. 10(c)] for the negative impedance converter including the varactor is from −2 to 0  at 1 GHz in bias conditions of VG of 0.5–0.7 V and V D of 0.8–2.3 V. The simulated stability factor of the whole coupler with the active circuit integrated (ports 1 and 3) is larger than 1 for all states, which means the circuit is absolutely stable. B. S-Parameter Response For experimental demonstration, an improved reconfigurable coupler with negative impedance converter operating at center frequency 1 GHz was designed and fabricated. Fig. 10(c) shows the final layout of the tunable coupler with negative impedance converter. Final dimensions of the coupler are as follows: W1 = 2.4, W2 = 0.94, W3 = 1.5, W4 = 1.5, L 1 = 43, L 2 = 9.8, L 3 = 7.5, g1 = 1, g2 = 0.3, g3 = 3.3, and g4 = 3.1 (all in millimeters). Figs. 12–14 show the simulated results of the improved tunable coupler with negative impedance converter. These results include the insertion losses, phase difference, return losses, and isolation versus different C D (0.7–10 pF). Fig. 12 shows the simulated insertion losses and phase differences at f 0 .

Fig. 13 shows that ideal return losses and port isolation are both realized at the center frequency of 1 GHz. The simulated bandwidth with return loss and port isolation of 15 dB is found to be about 10% for all states. Fig. 14 shows the comparison of the simulated insertion losses and phase differences versus power-dividing ratio at f 0 of the proposed tunable coupler with and without negative impedance converter. With the negative impedance converter, the simulated results show that the phase imbalance is less than 2° for all states. The insertion loss is close to the theoretical value. Figs. 15 and 16 show the measured S-parameters of the fabricated coupler with negative impedance converter for different power-dividing ratios. The measured results indicate that the tunable power-dividing output has been achieved around the center frequency of 980 MHz. The small discrepancies between the simulation and measurement are mainly caused by the dielectric constant tolerance of the substrate and the limited accuracy of circuit model of the negative impedance converters and varactors. Fig. 15 shows that the measured 15-dB return loss bandwidth is better than 80 MHz over the entire power-dividing ratio tuning range from about 24.4 to −22.2 dB (from 275.4 to 6E-4). The discrepancies between the design specification

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9

TABLE II P ERFORMANCE C OMPARISON OF THE R EPORTED Q UADRATURE T UNABLE C OUPLER

Fig. 17. Photograph of the fabricated tunable coupler with negative impedance converter.

without negative impedance converter. The measured phase imbalances at the center frequency are smaller than 1° for all states. The insertion loss is improved by 1 dB when compared with the one without active circuits. Fig. 17 shows an image of the fabricated tunable coupler with negative impedance converter. The coupler area is only 0.21λg × 0.1λg. Fig. 16. Comparison of the measured results versus power dividing ratio at the center frequency of the proposed tunable coupler with and without negative impedance converter. (a) −90° mode. (b) 90° mode. (Insets: close-up performance comparisons).

and measured power-dividing ratio range may be due to the parasitic inductance or capacitance from the negative impedance converter and bias circuit, which change the total capacitance tuning range. When tuning the voltage from 0 to 24 V, the insertion loss between ports 3 and 1 varies from 24.5 to 0.08 dB then to 1.02 dB (including connector loss). While the insertion loss between ports 4 and 1 varies from 0.06 to 22.3 dB then to 7.2 dB. The measured return loss and isolation are both better than 20 dB for all states at the center frequency. Fig. 16 shows the comparison of the measured insertion losses and phase differences versus powerdividing ratio at f0 of the proposed tunable coupler with and

C. Comparison Table II compares the performance of the tunable coupler in this paper with reported couplers with reconfigurable powerdividing ratio. Note that the couplers in [17] and [20] are made with RF MEMS and CMOS tunable active inductors, which increases the complexity and cost of the circuit fabrication. As shown, the presented tunable planar coupler of this paper offers the widest tuning range, a moderate coupler size, lowest insertion loss, and smallest phase imbalance. VI. C ONCLUSION This paper presented a compact planar continuously tunable coupler composed of a pair of transmission lines with the three fixed capacitors and two tunable capacitors. The closed-form design equations were derived by employing even–odd-mode analysis. For verification, a coupler operating at 1 GHz was

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demonstrated with power-dividing ratio from 16.2 to −35 dB (from 42 to 3.2E-4). By introducing the negative impedance converter in the coupler, the insertion loss was improved by 1 dB and phase imbalance was improved from 10° to 1°. The excellent performances including wide power-dividing tuning ratio, small size, low insertion loss, and small phase imbalance was demonstrated. ACKNOWLEDGMENT The author would like to thank the editors and reviewers of this paper for their valuable comments and suggestions. R EFERENCES [1] D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ, USA: Wiley, 2011. [2] F. Lin, Q.-X. Chu, and Z. Lin, “A novel tri-band branch-line coupler with three controllable operating frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 12, pp. 666–668, Dec. 2010. [3] L. Chiu and Q. Xue, “Investigation of a wideband 90° hybrid coupler with an arbitrary coupling level,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 4, pp. 1022–1029, Apr. 2010. [4] C.-L. Hsu, J.-T. Kuo, and C.-W. Chang, “Miniaturized dual-band hybrid couplers with arbitrary power division ratios,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 1, pp. 149–156, Jan. 2009. [5] X. Wang, W.-Y. Yin, and K.-L. Wu, “A dual-band coupled-line coupler with an arbitrary coupling coefficient,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 4, pp. 945–951, Apr. 2012. [6] H. L. Lee, D.-H. Park, M.-Q. Lee, and J.-W. Yu, “Reconfigurable 2×2 multi-port amplifier using switching mode hybrid matrices,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 2, pp. 129–131, Feb. 2014. [7] F. Ferrero, C. Luxey, R. Staraj, G. Jacquemod, M. Yedlin, and V. Fusco, “A novel quad-polarization agile patch antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1563–1567, May 2009. [8] Y.-S. Liu and J.-S. Row, “Back-to-back microstrip antenna fed with tunable power divider,” IEEE Trans. Antennas Propag., vol. 63, no. 5, pp. 2348–2353, May 2015. [9] E. E. Djoumessi, E. Marsan, C. Caloz, M. Chaker, and K. Wu, “Varactortuned dual-band quadrature hybrid coupler,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 11, pp. 603–605, Nov. 2006. [10] F. Ferrero, C. Luxey, R. Staraj, G. Jacquemod, and V. F. Fusco, “Compact quasi-lumped hybrid coupler tunable over large frequency band,” Electron. Lett., vol. 43, no. 19, pp. 1030–1031, Sep. 2007. [11] Y. Xiao, F. Lin, H. Ma, X. Tan, and H. Sun, “A planar balanced power divider with tunable power-dividing ratio,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 12, pp. 4871–4882, Dec. 2017. [12] S.-Y. Wang, D.-Y. Lai, and F.-C. Chen, “A low-profile switchable quadripolarization diversity aperture-coupled patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 522–524, 2009. [13] J.-S. Row and M.-J. Hou, “Design of polarization diversity patch antenna based on a compact reconfigurable feeding network,” IEEE Trans. Antennas Propag., vol. 62, no. 10, pp. 5349–5352, Oct. 2014.

[14] S. Y. Zheng, W. S. Chan, and Y. S. Wong, “Reconfigurable RF quadrature patch hybrid coupler,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3349–3359, Aug. 2013. [15] K.-K. M. Cheng and S. Yeung, “A novel rat-race coupler with tunable power dividing ratio, ideal port isolation, and return loss performance,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 55–60, Jan. 2013. [16] M. Zhou, J. Shao, B. Arigong, H. Ren, R. Zhou, and H. Zhang, “A varactor based 90° directional coupler with tunable coupling ratios and reconfigurable responses,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 3, pp. 416–421, Mar. 2014. [17] M. A. Y. Abdalla, K. Phang, and G. V. Eleftheriades, “A compact highly reconfigurable CMOS MMIC directional couplerr,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 2, pp. 305–319, Feb. 2008. [18] J. Sun, C. Li, Y. Geng, and P. Wang, “A highly reconfigurable lowpower CMOS directional coupler,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 9, pp. 2815–2822, Sep. 2012. [19] B. Hur and W. R. Eisenstadt, “Tunable broadband MMIC active directional coupler,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 168–176, Jan. 2013. [20] U. Shah, M. Sterner, and J. Oberhammer, “High-directivity MEMStunable directional couplers for 10–18-GHz broadband applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3236–3246, Sep. 2013. [21] Y.-H. Chun, J.-R. Lee, S.-W. Yun, and J.-K. Rhee, “Design of an RF low-noise bandpass filter using active capacitance circuit,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 2, pp. 687–695, Feb. 2005. [22] F. Lin, S. W. Wong, and Q.-X. Chu, “Compact design of planar continuously tunable crossover with two-section coupled lines,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 3, pp. 408–415, Mar. 2014. [23] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York, NY, USA: McGraw-Hill, 1992. [24] E. M. T. Jones and J. T. Bolljahn, “Coupled-strip-transmission-line filters and directional couplers,” IRE Trans. Microw. Theory Techn., vol. MTT-4, no. 2, pp. 75–81, Apr. 1956.

Feng Lin (M’14) received the B.Eng. degree in information engineering and Ph.D. degree in electromagnetic fields and microwave technology from the South China University of Technology, Guangzhou, China, in 2008 and 2013, respectively. His Ph.D. dissertation concerned design theory and realization of multiband couplers and power dividers. From 2013 to 2015, he was a Post-Doctoral Research Fellow with the University of Michigan, Ann Arbor, MI, USA. In 2016, he joined the School of Information and Electronics, Beijing Institute of Technology, Beijing, China, as an Associate Professor. His current research interests include RF microelectromechanical systems and millimeter-wave reconfigurable devices.

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Homodyne Digitally Assisted and Spurious-Free Mixerless Direct Carrier Modulator With High Carrier Leakage Suppression Weiwei Zhang, Graduate Student Member, IEEE, Abul Hasan, Graduate Student Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE, Mohamed Helaoui, Member, IEEE, Yongle Wu, Senior Member, IEEE, Lingxiao Jiao, and Yuanan Liu, Member, IEEE

Abstract— In this paper, a new method to design a digitally assisted and spurious-free direct carrier mixerless modulator based on the six-port correlator is proposed. The calibration of the modulator based on modified Cartesian memory polynomial (MCMP) is used to linearize and mitigate hardware impairment of the whole system. The modulation and the up conversion are performed by using the variable loads controlled by the differential in-phase and quadrature-phase baseband voltages together with common-mode voltages. The proposed MCMP is able to compensate for nonlinearity, frequency responses, residual carrier leakage, crosstalk between the in-phase and the quadrature-phase data. The proof-of-concept of digitally assisted mixerless modulator is developed and its performance is assessed at 2.6 GHz with modern communication signals. The error vector magnitudes between the input ideal baseband signals and the up-converted radio frequency signals are all between 2% and 4%. The residual carrier leakage, which remains present after imperfect suppression through hardware means, degrades the overall system performance and it can be suppressed completely by means of the proposed memory polynomial model. Index Terms— Carrier leakage, Cartesian memory polynomial, direct carrier modulator, memory effect, six-port correlator. Manuscript received April 26, 2017; revised June 22, 2017; accepted July 25, 2017. Date of publication August 15, 2017; date of current version March 5, 2018. This work was supported in part by the National Natural Science Foundations of China under Grant 61422103 and Grant 61671084, in part by the National Key Basic Research Program of China (973 Program) under Grant 2014CB339900, in part by BUPT Excellent Ph.D. Students Foundation under Grant CX2016303, in part by the China Scholarship Council, in part by the Alberta Innovates Technology Future, in part by the National Science and Engineering Research Council of Canada, and in part by the Canada Research Chairs Program. (Corresponding author: Weiwei Zhang.) W. Zhang is with the Beijing Key Laboratory of Work Safety Intelligent Monitoring, School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China, and also with the Intelligent RF Radio Laboratory, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: [email protected]). A. Hasan, F. M. Ghannouchi, and M. Helaoui are with the Intelligent RF Radio Laboratory, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Y. Wu, L. Jiao, and Y. Liu are with the Beijing Key Laboratory of Work Safety Intelligent Monitoring, School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2735408

I. I NTRODUCTION INCE the six-port correlator was used for direct digital receiver for retrieving the phase shift keying (PSK) and quadrature amplitude modulation (QAM) baseband signals for the first time in [1], the homodyne structures based on the multiport techniques as radio frequency (RF) front-end, have attracted more attention. The RF front-ends using multiport techniques can be classified into multiport demodulators [2]–[8] and multiport modulators [9]–[30]. Different from the multiport demodulator, the multiport modulator, which is shown in Fig. 1(a), can realize the function of frequency up-conversion from baseband signal to RF signal without a mixer and hence it can replace the traditional direct conversion modulator based on two mixers and one phase shifter, as shown in Fig. 1(b) in the modern wireless communication systems due to the easy fabrication, low-cost, low-power consumption, and wideband coverage. As shown in Fig. 1(a), the direct carrier mixerless modulator essentially consists of one six-port correlator and variable loads, which are controlled by the baseband signals to generate different reflection coefficients for RF modulation. Hence, the optimization and design of variable loads are very important as they directly affect the overall system performance. Many six-port modulators (SPMs) with different circuit structures [9]–[30] have been proposed since [9]. There are mainly three different methods to realize the variable loads, namely, RF switches [9]–[17], diodes [18]–[22], as well as transistors [23]–[30]. However, there are still many problems in the SPMs which deteriorate the system performance that need to be solved, and some of them are listed here. First, carrier leakage exists in almost all of the SPMs reported in the literature [12], [16], [18], [21], [26], and [28], which degrades the system performance and lowers the error vector magnitude (EVM). The carrier leakage is very prominent in [16] that uses only four ports for implementation and measurement. The method based on phase shifters together with differential baseband signal is adopted in [18]–[21], [26], [28], and [29], however, the carrier leakage cannot be suppressed completely from the modulated RF signal if the diodes are not identical, the electrical length of the transmission line implementing 90° phase shift is not

S

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Fig. 1.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 3, MARCH 2018

System configurations of direct carrier modulator based on (a) multiport correlator and (b) two mixers.

optimal at the operating frequency, or the six-port correlator is not ideal. Second, the crosstalk between the in-phase and quadrature-phase (I/Q) channels is another factor that influences the system performance, and it mainly results from the nonideal characteristic of the six-port correlator, or the variable loads. It can be seen from [19] and [24] that reflection coefficients of the variable loads are complex when the loads are realized by the diodes and the transistors which will lead to the I/Q signal imbalance. Third, RF switches [9]–[17] are limited to low switching rate, and some specific loads, such as open and short terminations, are required for the quadrature phase shift keying (QPSK) [9], [12]–[17] and QAM [11], [13] signals modulation. Though the diodes [18] and the transistors [24] feature high-speed performance for the SPM, they exhibit nonlinearity and memory effects similar to power amplifiers (PAs) [31]–[34] which negatively impact the SPM performance. Notably, the architecture [29], [30] avoids the nonlinearity issue of the reflection coefficients from the impedance loads in that only binary baseband data are utilized, however, the increased number of the power dividers (PDs) and transistors make the circuit very complicated. Finally, only QPSK and QAM signals have been tested in the SPMs [9]–[30]. The whole system performance for practical communication signals such as wideband code division multiple access (WCDMA) signal, and long term evolution (LTE) signal have not been used before to assess the suitability of such modulator/up-convertor in realistic modern conditions. On the other hand, the digital predistortion (DPD) model based on the memory polynomial [35]–[40], which is an effective method to linearize the PAs, has been successfully used in behavioral modeling of the PAs to decrease adjacent channel leakage ratio and improve linearity as well as efficiency. System imperfections and hardware impairments can be compensated greatly when the DPD method is adopted in that it can mimic inverse characteristic of the PAs. However, the DPD method has not been used in the SPMs [9]–[30] based transmitting systems despite its architectural similarity with the DPD-PA system. Therefore, a method based on the modified Cartesian memory polynomial (MCMP) is proposed in this paper to realize a direct carrier mixerless SPM with high performance that features: 1) high linearity and relatively very good EVMs between 2% and 4%; 2) complete carrier leakage suppression;

Fig. 2.

System configuration of an SPM based on voltage variable loads.

3) the nonlinearity and the memory effect of the diodes being taken into consideration in the mixerless SPM for the first time; 4) modern communication signals, e.g., WCDMA and LTE signals, used in the SPM systems evaluation. The remainders of this paper are organized as follows. Section II mainly discusses the circuit structure of the SPM, the baseband control circuit, the six-port correlator, and some associated problems. Then, the DPD model based on the MCMP is proposed and analyzed in Section III. A complete test setup and a simple measurement procedure are provided in Section IV. Furthermore, the commonly used modulated signals are used to test the proposed mixerless modulator in Section V. Besides, the discussion about the MCMP, and comparison with other state-of-the-art SPMs are also presented. Finally, a conclusion is drawn in Section VI. II. C IRCUIT A NALYSIS OF THE S IX -P ORT M ODULATOR AND P ROBLEM D ISCUSSION Schematic of a typical SPM [18] is shown in Fig. 2, which essentially consists of one PD, three quadrature couplers (QCs), two transmission lines Z 0 with the electrical length θ0 at the operating frequency f 0 , and four variable loads Z k (k = 3, . . . , and 6). The local oscillator (LO) signal aLO (t), which goes into port 1 on the left-hand side, experiences different paths to the kth port of the six-port correlator, and then gets reflected because of impedance mismatch.

ZHANG et al.: HOMODYNE DIGITALLY ASSISTED AND SPURIOUS-FREE MIXERLESS DIRECT CARRIER MODULATOR

Fig. 3.

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(a) Schematic of the baseband control circuit and (b) manufactured baseband control circuit board (units: mm).

Finally, parts of the reflected signal, which contain the I/Q data information, are combined at port 2 to generate the desired modulated RF signal. It is worth noting that the variable loads Z k are all controlled by the baseband I/Q signals, which play a key role in the realization of the direct carrier modulator. A theoretical analysis about ideal operation of the SPM has been discussed in [18], however, the measured results reported therein leave further scope of improvements, which cannot be avoided in any practical implementation. Hence, a generalized circuit theory for SPM is provided and the existing problems are discussed in this section. An efficient method to address some of these challenges is proposed in Section III to improve the system performance. Fig. 4. Measured reflection coefficients (3 and 4 ) versus the dc voltages.

A. Baseband Control Circuit Analysis The Schottky diodes HSM286Y from Broadcom Limited are adopted in this paper due to their high speed and low complexity [18]. Besides, the cost of the diodes and their power consumptions are much lower than those of the transistors [23]–[30]. The baseband control circuit consisting of a quarter-wavelength transmission line Z 0 , the Schottky diodes, and a bias circuit is shown in Fig. 3(a). A manufactured baseband control circuit board with optimized dimensions is illustrated in Fig. 3(b). It can be seen that the differential baseband signal together with the bias common-mode (CM) voltage (Vk ) is used to change the input impedances of the Schottky diodes. Capacitors C1 and C2 are used as RF choke to prevent the LO signal from going into the direct current (dc) source, and to filter the harmonic signals from the voltages Vk , simultaneously. In addition, the quarter-wavelength transmission lines Z 1 are used to block the LO signal, which comes from ports k(k = 3, 4, . . . , 6). The capacitor C3 is used to prevent the voltage Vk from flowing into the six-port correlator. Here, the parameters are chosen as: Z 1 = Z 0 = 50 , θ0 = 90°, C1 = 220 pF, C2 = 1000 pF, and C3 = 470 pF. Then, the reflection coefficient (k ) can be measured by using a vector network analyzer N5230A from the Keysight Technologies under different dc voltages. Since the baseband

control circuits are identical, only the top part at ports 3 and 4 for the in-phase signal in Fig. 2 is discussed, and the corresponding measured results (3 and 4 ) are plotted in Fig. 4. It can be seen that the complex reflection coefficients (V ) are not linear from 0 to 400 mV, and only quasi-linear in the range of 100 to 200 mV. The complex reflection coefficient (V ) can be expanded by Taylor series at V0 , and we can derive that (V ) = (V0 ) +  (1) (V0 ) · (V − V0 ) + +

1 (2)  (V0 ) · (V − V0 )2 2!

1 (3) 1  (V0 ) · (V − V0 )3 +  (4) (V0 ) · (V − V0 )4 . . . 3! 4! (1)

where (V0 ) is the reflection coefficient (V ) at voltage V0 ,  (m) (V0 ) is the mth derivative of (V ) at V0 , and m is a positive integer. Notably, the coefficients ((V0 ),  (1) (V0 ), . . . ,  (m) (V0 ), . . .) in (1) are all complex. In this paper, the differential baseband voltages (v I or v Q ) plus the CM voltage (Vcm ) [18] are

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used to control the Schottky diodes, namely, V3 = Vcm + v I V4 = Vcm − v I .

(2a) (2b)

It is worth noting that the baseband control circuits at ports 3 and 4 and ports 5 and 6 have been assumed identical, and we can get that

Then we can obtain the reflection coefficients 3 and 4 after substituting (2) into (1) if we assume V = V3 (V4 ), and V0 = Vcm (1)

3 (V3 ) = 3 (Vcm ) + 3 (Vcm ) · v I +

1 (2)  (Vcm ) · v 2I 2! 3

1 (3) 1 (4) + 3 (Vcm ) · v 3I + 3 (Vcm ) · v 4I . . . 3! 4! (3a) 1 4 (V4 ) = 4 (Vcm ) − 4(1) (Vcm ) · v I + 4(2) (Vcm ) · v 2I 2! 1 (3) 1 (4) 3 − 4 (Vcm ) · v I + 4 (Vcm ) · v 4I . . . . 3! 4! (3b) If the quasi-linear part of the reflection coefficient in Fig. 4 is used (Assumption 1), the high-order terms (higher than first-order) can be omitted, and (3) can be simplified into 3 (V3 ) = 3 (Vcm ) + 3(1) (Vcm ) · v I

(4a)

4 (V4 ) =

(4b)

(1) 4 (Vcm ) − 4 (Vcm )

· v I .

The reflection coefficients 3 (V3 ) and 4 (V4 ) can be viewed symmetrical along the middle line ( = 0) in Fig. 4 if we assume all Schottky diodes are identical and the electrical length θ0 of the transmission line Z 0 is 90° at the operating frequency (Assumption 2). Then, we can get (5) under Assumptions 1 and 2 4 (Vcm ) = −3 (Vcm )

(1) 4 (Vcm )

=

(5a)

(1) −3 (Vcm ).

(5b)

After combining (4) and (5), the reflection coefficients 3 (V3 ) and 4 (V4 ) can be written as 3 (V3 ) = 3 (Vcm ) + 3(1) (Vcm ) · v I

(6a)

4 (V4 ) = −3 (Vcm ) +

(6b)

(1) 3 (Vcm ) · v I .

The same analysis can also be extended to the baseband control circuits at ports 5 and 6, and we can derive that 5 (V5 ) = 6 (V6 ) =

(1) 5 (Vcm ) + 5 (Vcm ) (1) 6 (Vcm ) − 6 (Vcm )

· v Q

(7a)

· v Q .

(7b)

Similarly, (8) can be satisfied under Assumptions 1 and 2 6 (Vcm ) = −5 (Vcm ) 6(1) (Vcm ) = −5(1) (Vcm ).

(8a) (8b)

Finally, the reflection coefficients 5 (V5 ) and 6 (V6 ) can be written as 5 (V5 ) = 5 (Vcm ) + 5(1) (Vcm ) · v Q

(9a)

6 (V6 ) = −5 (Vcm ) +

(9b)

(1) 5 (Vcm ) · v Q .

3 (Vcm ) = 5 (Vcm )

(10a)

3 (Vcm ) = 5 (Vcm ).

(10b)

(1)

(1)

B. Analysis of the Six-Port Correlator On the other hand, the total output waves at port 2 [18] can be calculated as aRF (t) = aLO (t)

6 

Sk1 k S2k

k=3

= aLO (t)(S31 3 S23 + S41 4 S24 + S51 5 S25 + S61 6 S26 ). (11) There are four different terms in (11), and each term is generated by the following three steps [18]. 1) The LO signal is transferred to port k. 2) The signal at port k is reflected by the variable loads Z k . 3) All the reflected signals are combined at port 2. Assuming that the six-port correlator is ideal (Assumption 3), its scattering parameters can be obtained as ⎤ ⎡ S11 S12 S13 S14 S15 S16 ⎢ S21 S22 S23 S24 S25 S26 ⎥ ⎥ ⎢ ⎢ S31 S32 S33 S34 S35 S36 ⎥ ⎥ [S] = ⎢ ⎢ S41 S42 S43 S44 S45 S46 ⎥ ⎥ ⎢ ⎣ S51 S52 S53 S54 S55 S56 ⎦ S61 S62 S63 S64 S65 S66 ⎤ ⎡ 0 0 1 j j −1 ⎢ 0 0 −1 j −1 j ⎥ ⎥ ⎢ ⎥ 1⎢ 1 −1 0 0 0 0 ⎥. ⎢ (12) = ⎢ ⎥ j j 0 0 0 0 2⎢ ⎥ ⎣ j −1 0 0 0 0 ⎦ −1 j 0 0 0 0 After substituting (12) into (11), we can get 1 aRF (t) = − aLO (t)[(3 + 4 ) + j (5 + 6 )]. 4

(13)

Then the modulated RF signal can be obtained after combining (6) and (9)–(11)

(1) 1 (1) aRF (t) = − aLO (t) 3 (Vcm ) · v I + j 5 (Vcm ) · v Q 2 1 (1) (14) = − aLO (t)3 (Vcm )(v I + j v Q ). 2 It can be seen that the first term in (14) is the modulating in-phase signal, and the second term is the modulating quadrature-phase signal. Therefore, this SPM can realize frequency up-conversion from the baseband to RF if the Assumptions 1–3 are all satisfied.

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C. Problems Discussion For any practical implementation of the SPM system, the ideal conditions leading to (14) cannot be guaranteed due to many factors which cannot be omitted, and some of which are discussed here. 1) The reflection coefficient (V ) is not linear, which can be seen from Fig. 4. Hence, there are many highorder terms (higher than first-order) generated as listed in (3a) and (3b), leading to spectrum regrowth and signal distortion in the modulated RF signal at port 2. 2) The reflection coefficient (V ) is only quasi-linear from 100 to 200 mV, which limits the range of v I (v Q ) leading to worse performance especially for modulated signal with high peak to average power ratio (PAPR), such as WCDMA signal and LTE signal. 3) The memory effects of the Schottky diodes will affect the overall system performance which must also be compensated in the system modeling. 4) If the electrical length θ0 of the transmission line Z 0 deviates from 90° at the operating frequency, or the Schottky diodes are not identical, namely, the actual reflection coefficients (3 (V3 ) and 4 (V4 ) or 5 (V5 ) and 6 (V6 )) are not symmetrical along the middle line ( = 0), (15) can be derived after combining (4) and (7) under Assumptions 1 and 3 3 + 4 = (3 (Vcm ) + 4 (Vcm ))   Carrier Leakage

(1)

(1) + 3 (Vcm ) − 4 (Vcm ) · v I  

(15a)

Fundamental Signal

5 + 6 = (5 (Vcm ) + 6 (Vcm ))   Carrier Leakage

(1)

(1) + 5 (Vcm ) − 6 (Vcm ) · v Q . (15b)  

Fig. 5.

Block diagrams of the (a) SPM and (b) DPD-based linear SPM.

It can be observed from (16) that the carrier leakage exists in the modulated RF signal, and the crosstalk phenomenon arises because the complex coefficients of the v I and v Q are not equal. From the above analysis, we can conclude that the carrier leakage, I/Q signals crosstalk, as well as the nonlinearity and the memory effects of the diodes cannot be removed after taking all the practical conditions into consideration, and they deteriorate the overall system performance. III. L INEARIZATION OF THE S IX -P ORT M ODULATOR Based on the theoretical analysis of the SPM in Section II, the MCMP, which can be viewed as an inverse model of the SPM, is proposed to make the whole system quasi-linear in this section. Therefore, the carrier leakage, the I/Q signals crosstalk, the nonideal characteristic of the six-port correlator, as well as the nonlinearity and the memory effects of the diodes can be taken into consideration if the DPD technique is applied in the SPM system. A. Six-Port Modulator Model As shown in Fig. 5(a), the input baseband signal x in (n) and its corresponding signal yout (n) from output RF signal can be expressed as

Fundamental Signal

It can be seen that the first term in (15) contributes to the carrier leakage, which will have a negative impact on the modulator performance. The complex coefficients of the second term in (15a) and (15b) are not equal, which brings about the crosstalk phenomenon between the in-phase and the quadrature-phase components of the signal. 5) The nonideal characteristic of the six-port correlator also results in the carrier leakage and the I/Q signal crosstalk. Equation (16), shown at the bottom of the page, can be obtained after substituting (6), (9) into (11) under Assumptions 1 and 2.

x in (n) = x I (n) + j x Q (n)

(17a)

yout (n) = y I (n) + j y Q (n).

(17b)

The MCMP model of the SPM is proposed in (18) according to the theoretical analysis of the direct carrier modulator in Section II, namely, y I (n) = α0 +

k  M N  

k− p

αkpm x I

p

(n − m) · x Q (n − m)

k=1 p=0 m=0

(18a) y Q (n) = β0 +

k  M N  

k− p

βkpm x I

p

(n − m) · x Q (n − m).

k=1 p=0 m=0

(18b)

(1) (1) aRF (t) = aLO (t) S31 S23 (3 (Vcm ) + 3 (Vcm ) · v I ) + S41 S24 (−3 (Vcm ) + 3 (Vcm ) · v I ) (1)

(1)

+ S51 S25 (5 (Vcm ) + 5 (Vcm ) · v Q ) + S61 S26 (−5 (Vcm ) + 5 (Vcm ) · v Q ) = aLO (t) S31 S23 3 (Vcm ) − S41 S24 3 (Vcm ) + S51 S25 5 (Vcm ) − S61 S26 5 (Vcm )   + (S31 S23 + 

Carrier Leakage (1) S41 S24 )3 (Vcm ) · v I + (S51 S25



Fundamental Signals





+ S61 S26 )5(1) (Vcm ) · v Q  Fundamental Signals





(16)

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where parameter N denotes the nonlinearity order, M is the memory depth, and coefficients (α0 , β0 , αkpm , and βkpm ) are all real-valued system constants. Notably, the unknown parameters α0 and β0 represent the carrier leakage. Actually, the output signal yout (n) can be written in a more compact form after combining (17b) and (18), namely, yout (n) = δ0 +

k  M N  

k− p

δkpm x I

p

(n − m) · x Q (n − m)

k=1 p=0 m=0

(19) where δ0 = α0 + jβ0

Fig. 6.

Block diagram of the proposed DPD model for SPM.

Fig. 7.

Complete test setup for the SPM.

(20a)

δkpm = αkpm + jβkpm .

(20b)

B. Nonlinearity Analysis of the Six-Port Modulator Model If we assume M = 0 and N = 3, the output signal of the SPM in (19) can be simplified into a memoryless model, which is listed in the following: yout (n) = δ0   Carrier Leakage

+ δ100 x I (n) + δ200 x I2 (n) + δ300 x I3 (n)  

+δ110 x Q (n) + 

In-Phase Signal and its Higher-Order Terms 2 3 δ220 x Q (n) + δ330 x Q (n)





Quadrature Signal and its Higher-Order Terms + δ210 x I (n)x Q (n)+δ310 x I2 (n) · x Q (n)+δ320 x I (n)





2 · xQ (n) . 

Cross-Products of the In-Phase and Quadrature Signals

(21) It can be seen from (21) that this expression contains not only the carrier leakage, the I/Q signals, and their high-order terms, but also the high-order cross products of the I/Q signals. Hence, this model can approximate the behavior modeling of the SPM in Fig. 5(a). C. DPD Model for Linearization The inverse model of the SPM, namely, the DPD model as shown in Fig. 5(b), can be easily derived from (19) by using the similar form u out_DPD (n) = ε0 +

M N  k  

k− p

εkpm w I

(n − m)

k=1 p=0 m=0 p · w Q (n

− m)

(22)

where win_DPD (n) = w I (n) + j w Q (n) is the input signal of the DPD model, the unknown coefficients (ε0 and εkpm ) are all complex, and the model coefficient ε0 is used to suppress the carrier leakage. Equation (22) can be written in a matrix form as shown in the following: → −−−→ − u out_DPD (n) = E · W (n)

(23)

where − → E = [ε0 ε100ε101 . . . εkpm . . . εNNM ] ⎡ ⎤ 1 ⎢ ⎥ w I (n) ⎢ ⎥ ⎢ ⎥ w I (n − 1) ⎢ ⎥ ⎢ ⎥ .. −−−→ ⎢ ⎥ . W (n) = ⎢ ⎥. p ⎢ k− p ⎥ ⎢ w I (n − m)w Q (n − m) ⎥ ⎢ ⎥ .. ⎢ ⎥ ⎣ ⎦ . N (n − M) wQ

(24a)

(24b)

In addition, the block diagram of the DPD model shown in Fig. 6 can be seen as a combination of one constant ε0 , which is used to suppress the carrier leakage, and (M + 1) polynomial functions, each of which is applied to a different time delayed version of the input signal. IV. C OMPLETE T EST S ETUP AND M EASUREMENT P ROCEDURE The complete measurement setup for the SPM operating at 2.6 GHz is illustrated in Fig. 7. The original baseband signals, such as QAM signal, WCDMA signal, and LTE signal, are downloaded into the arbitrary waveform generator (AWG) 81180A through the ethernet cable, and the outputs are the baseband differential I /Q signals (v I and v Q ) plus the CM voltages (Vcm ). These four voltage signals (Vcm ± v I and Vcm ± v Q ) are used to control the Schottky diodes to change their impedances. The LO signal from the vector signal

ZHANG et al.: HOMODYNE DIGITALLY ASSISTED AND SPURIOUS-FREE MIXERLESS DIRECT CARRIER MODULATOR

Fig. 8. (a) Constellations and (b) power spectra of a 16-QAM signal with 4-MHz bandwidth without using the DPD model.

generator (VSG) N5182A is fed into the six-port correlator. Then, the output modulated RF signal generated from the modulator is fed into a mixed signal oscilloscope (MSO) 9404A, and captured by using vector signal analyzer (VSA) 89600. The AWG 81180A, VSG N5182A, MSO 9404A, and the VSA 89600 are all from the Keysight Technologies. The peak-to-peak voltages of the differential I/Q signals is set to be 150 mV, the CM voltage Vcm is 100 mV, and the amplitude of the LO signal is set to be −10 dBm in all the measurements in this paper. Furthermore, the measurement procedure is given here for better understanding of the system. 1) Download the original baseband signal x in (n) into AWG 81180A, and capture the corresponding output signal yout (n) by using the measurement setup as shown in Fig. 7. 2) Determine the unknown coefficients of the DPD model in (22) in which the input as well as the output signals are win_DPD (n) = yout (n)/G u out_DPD (n) = x in (n)

(25a) (25b)

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Fig. 9. Power spectra of (a) WCDMA signal with 3.84-MHz bandwidth and (b) LTE signal with 1.4-MHz bandwidth without using the DPD model.

where G is the conversion gain of the SPM, and it should be less than 1, which indicates that there is no signal amplification in the system. The first 1000 data symbols are enough in this paper to calculate the unknown coefficients in (22). Apparently, the nonlinearity order (N) and the memory depth (M) should be determined in advance. 3) Obtain the new input signal of the SPM u out_DPD (n) by applying the original baseband signal x in (n) to the DPD model, namely, win_DPD (n) = x in (n).

(26)

4) Download the new baseband signal u out_DPD (n) into AWG 81180A, and capture the final output signal z out (n) from the modulated RF signal. 5) Calculated the EVM in (27) by comparing the final output signal from the RF signal with the original baseband signal. The final output signals should be normalized to the conversion gain of the SPM for amplitude scaling, and rotated by

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Fig. 10. (a) Signal constellations and (b) power spectra of a 16-QAM signal with 4-MHz bandwidth based on the proposed DPD model.

a certain angle to align its phase with the original signal   L 1  L |Z (n) − x in (n)|2  n=1 out  × 100%. (27) EVM =  L  1  2 |x in (n)| L n=1

Here L is the total length of the baseband signal. V. R ESULTS AND D ISCUSSION Some of the concerns of the conventional SPM have been discussed in Section II, and an effective method for digital compensation based on the MCMP model is proposed in Sections III and IV. Here, the common modulated signals, such as QAM signal, WCDMA signal, and LTE signal, are used to validate the proposed method in this section. A. Measured Results Without DPD The measured results without applying the DPD model to the SPM are discussed first to explain why the system performance can be improved greatly after taking the proposed MCMP model into consideration.

Fig. 11. Power spectra comparison of (a) WCDMA signal with 3.84-MHz bandwidth and (b) LTE signal with 1.4-MHz bandwidth based on the proposed DPD model.

The 20 000 random symbols for 16-QAM signal with 4-MHz bandwidth are generated by matrix laboratory (MATLAB) software from the MathWorks. The sampling frequency is 80 MHz, and its PAPR is 6.7622 dB. Only step (1) in Section IV is used to obtain the output signal. Fig. 8(a) shows the constellations of the 16-QAM signal, and the corresponding power spectra are plotted in Fig. 8(b). Notably, the carrier frequency of the original baseband signal is set to be 2.6 GHz for better comparison. The calculated EVM is 13.2%, and the symbols of the output signal in Fig. 8(a) deviate from their ideal desired positions resulting in very bad constellation and system performance. Besides, 20 491 random symbols of a WCDMA signal with 3.84-MHz bandwidth, and 15 360 random symbols of a LTE signal with 1.4-MHz bandwidth are also tested. Their sampling frequencies are 61.44 and 7.68 MHz, respectively. In addition, the PAPR is 9.5155 dB (8.9325 dB) for WCDMA (LTE) signal. The power spectra are plotted in Fig. 9, and the EVM is 20.2% (14.5%) for the WCDMA (LTE) signal. It can be seen that the measured results are very bad, and the EVMs are all larger than 10%. The carrier leakage has

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TABLE I S UMMARY OF THE M EASUREMENT R ESULTS

Fig. 12. Constellations of a 64-QAM signal with 10-MHz bandwidth based on (a) DPD-1 and (b) DPD-2. The power spectra of the 64-QAM signal with 10-MHz bandwidth based on (c) DPD-1 and (d) DPD-2.

been suppressed for the 16-QAM signal in Fig. 8(b), however, it still exists in Fig. 9, which shows that the method [18] based only on the phase shifters and the differential baseband

voltages cannot remove the carrier leakage completely for the WCDMA and LTE signals. Apart from the carrier leakages, the I/Q crosstalk, as well as the nonlinearity and the memory

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effects of the diodes deteriorate the system performance, which lead to unacceptable system performance. B. Measured Results Based on the Proposed DPD Model In order to verify that the proposed DPD model can address all the concerns raised here and can improve the system performance greatly, the same signals are also measured by using the test procedure described in Section IV. The constellation and the power spectra of the 16-QAM signal with 4-MHz bandwidth are plotted in Fig. 10, and the power spectra of the WCDMA as well as LTE signals are illustrated in Fig. 11(a) and (b), respectively. The calculated EVM for the 16-QAM is only 2.7% when N = 3 and M = 2, which is improved greatly when compared with the results (13.2%) without the DPD model. As demonstrated by Fig. 10(a), the input (original) signal and the output (modulated RF) signal match very well. In addition, the EVM is 3.9% (3.7%) for the WCDMA (LTE) signal when N is 3 (4), and M is 9 (2). It can be seen that the system performance, in terms of EVM, improves greatly when the proposed DPD model is applied during the measurement. More importantly, the residual carrier leakage, which still exists in Fig. 9, is removed completely, which can be seen in Fig. 11. Besides the modulated signals discussed above, other baseband signals with different bandwidth and PAPR are also tested by using the method proposed in this paper, and the final results are summarized in Table I. It can be seen that the EVMs for all the test conditions are between 2% and 4%, indicating the validity and the performance of the proposed methods for SPM. Fig. 13. Power spectra of the LTE signal with 5-MHz bandwidth obtained based on (a) DPD-1 and (b) DPD-3.

C. Discussion About the I/Q Signal Crosstalk It can be seen that the DPD model in (22) consists of I/Q cross products, which may be caused by the six-port correlator due to the nonideal isolation among four ports k. Therefore, another DPD model (DPD-2), which only consists of the I/Q signals, their higher-order terms and the past values, is proposed in (28) for comparison. The original DPD model in (22) is named as DPD-1 for better explanation u out_DPD (n) = ε0 + +

M N  

k=1 m=0 M N 

ε1km wkI (n − m)

ε2km wkQ (n − m).

(28)

k=1 m=0

The 20 000 random symbols of a 64-QAM signal with 10-MHz bandwidth are tested by applying these two DPD models, namely, DPD-1 in (22) and DPD-2 in (28). The constellations and the corresponding power spectra are illustrated in Fig. 12 for comparison. The final EVMs are 2.9% and 4.1%, respectively, for these two DPD models under the same nonlinearity order and the memory depth (N = 3 and M = 9). The performance improves when the cross-products of the baseband signals are taken into consideration, and the constellation in Fig. 12(a) is better than that in Fig. 12(b).

Fig. 14. Power spectra of the LTE signal with 5-MHz bandwidth from 2.5998 to 2.6002 GHz.

D. Discussion About the Carrier Leakage Since carrier leakage has a negative impact on the final system performance, the measured results based on a DPD model (DPD-3) in (29) are compared with those based on the DPD-1 model in (22) to verify that the coefficient ε0 is used to suppress the carrier leakage.

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TABLE II C OMPARISON W ITH THE S TATE - OF - THE -A RT R ESULTS

Fig. 15. Complete set-up based on the signal generator from Keysight Technologies as a RF modulator.

The DPD-3 is written in (29), and the only difference from (22) is that there is no complex coefficient ε0 in the following: u out_DPD (n) =

k  M N  

k− p

εkpm w I

p

(n − m) · w Q (n − m).

k=1 p=0 m=0

(29) The 15 360 random symbols of the LTE signal with 5-MHz bandwidth are tested and their spectra from 2.592 and 2.608 GHz are plotted in Fig. 13. The calculated EVMs are 2.6% (15.4%) based on the DPD-1 (DPD-3) under the same conditions (N = 4 and M = 2), and the overall system performance improves greatly. There is no carrier leakage in Fig. 13(a) for the measured results based on DPD-1, however, it exists in Fig. 13(b), which leads to the degraded system performance. Besides, the spectrum regrowth in Fig. 13(a) is lower than that in Fig. 11(b). It can be fully verified that the coefficient ε0 plays a key role in suppressing the carrier leakage from Fig. 13. Furthermore, the results about the power spectra from 2.5998 to 2.6001 GHz are also plotted in Fig. 14 for clarity. It can be seen that the output signal obtained based on the DPD-1 remain consistent with the original input signal, and no carrier leakage at 2.6 GHz exists. E. Comparison With Signal Generator From Keysight Technologies A comparison with the results obtained by using the VSG N5281A as the RF signal modulator is also made here, and the corresponding measured setup is shown in Fig. 15. It is well known that the VSG N5182A can generate high-performance modulated RF signals. The original signal is downloaded into the VSG N5182A through the ethernet cable with preset amplitude (−10 dBm) and frequency (2.6 GHz). The output RF signal goes into one of the ports of the MSO9404A, and captured by using the VSA 89 600. It can be seen that the MSO9404A and the software VSA 89 600 are both used for capturing the final RF signal. The 40 000 random symbols of the 16-QAM signal with 1 MHz are tested by using the two setups in Fig. 7

(DPD-SPM system) and Fig. 15 (VSG), respectively. The constellation and the corresponding spectra are plotted in Fig. 16. The EVM is 3.2% based on the DPD-SPM system, and it is only 3.1% by using the VSG N5182A, which confirm that the SPM together with the proposed DPD model has the same function as the VSG though the measured results obtained from the VSG is slightly better than those obtained using the proposed DPD-SPM system. Notably, the constellation in Fig. 16(b) is mainly affected by the noise introduced in the system, and the MSO9404A. The similar influence from the noise also affects the performance of the proposed DPD-SPM system. F. Comparison With State-of-the-Art Six-Port Modulators The performance of state-of-the-art SPMs and our work are summarized in Table II. Besides the better performance compared with other results, the SPM system with real communication signals such as WCDMA and LTE signals are reported in this paper with improved system performance. In addition, the method based on conventional SPM together with the DPD model is proposed and tested for the first time. It is worth noting that the proposed MCMP model can be used for not only the Schottky diodes [18]–[22], but also

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Fig. 16. Constellation of the 16-QAM signal with 1-MHz bandwidth captured by using the (a) DPD-SPM system and (b) VSG. The power spectra of the 16-QAM signal captured by using the (c) DPD-SPM system and (d) VSG.

for the transistors [23]–[30] because they possess the similar characteristics in terms of nonlinearity and memory effects as discussed in Section II. VI. C ONCLUSION A homodyne mixerless and quasi-linear transmitter architecture is designed and implemented in this paper. The proposed architecture is based on the combination of an SPM and a DPD model to suppress the residual carrier leakage and to compensate for system nonidealities. It improves the overall system performance. The basic theory of the SPM and its problems including the carrier leakage, the crosstalk between the in-phase and quadrature signals, nonlinearity and memory effect of the Schottky diodes are clearly discussed. Then, the MCMP model, which consists of not only the present signals as well as their past samples and nonlinear terms, but also the cross-products and the carrier leakage, is proposed to make the whole system quasi-linear. Common modulated signals with different bandwidths and PAPRs are tested to validate the proposed system, and the final EVMs are between 2% and 4%. The measurement results confirm the dual approach of hardware and software based methods for very high carrier leakage suppression in direct carrier SPMs. Digitally assisted SPM

based on the proposed MCMP model would be a low-cost and power efficient solution for RF front-end designs for frequency up-conversion systems. ACKNOWLEDGMENT The authors would like to thank A. Kwan and A. Abdelhafiz, both with the Intelligent RF Radio Laboratory, University of Calgary, Calgary, AB, Canada, for their technical support and suggestions during the measurement. R EFERENCES [1] J. Li, R. G. Bosisio, and K. Wu, “A six-port direct digital millimeter wave receiver,” in IEEE MTT-S Int. Microw. Symp. Dig., San Diego, CA, USA, May 1994, pp. 1659–1662. [2] P. Perez-Lara, I. Molina-Fernandez, J. G. Wanguemert-Perez, and R. G. Bosisio, “Effects of hardware imperfection on six-port direct digital receivers calibrated with three and four signal standards,” Proc. Inst. Elect. Eng.—Microw., Antennas Propag., vol. 153, no. 2, pp. 171–176, Apr. 2006. [3] J. Osth et al., “Six-port gigabit demodulator,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 1, pp. 125–131, Jan. 2011. [4] C. de la Morena-Álvarez-Palencia and M. Burgos-Garcia, “Fouroctave six-port receiver and its calibration for broadband communications and software defined radios,” Prog. Electromagn. Res., vol. 116, pp. 1–21, 2011. [Online]. Available: http://www.jpier.org/ PIER/pier116/01.11030407.pdf

ZHANG et al.: HOMODYNE DIGITALLY ASSISTED AND SPURIOUS-FREE MIXERLESS DIRECT CARRIER MODULATOR

[5] A. Hasan and M. Helaoui, “Novel modeling and calibration approach for multi-port receivers mitigating system imperfections and hardware impairments,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 8, pp. 2644–2653, Aug. 2012. [6] A. Hasan and M. Helaoui, “Performance driven six-port receiver and its advantages over low-IF receiver architecture,” J. Electr. Comput. Eng., vol. 2014, Jan. 2014, Art. no. 198120. [7] S. O. Tatu, A. Serban, M. Helaoui, and A. Koelpin, “Multiport technology: The new rise of an old concept,” IEEE Microw. Mag., vol. 15, no. 7, pp. S34–S44, Nov./Dec. 2014. [8] W. Zhang, A. Hasan, F. M. Ghannouchi, M. Helaoui, Y. Wu, and Y. Liu, “Novel calibration algorithm of multiport wideband receivers based on real-valued time-delay neural networks,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 11, pp. 3540–3548, Nov. 2016. [9] Y. Zhao, C. Viereck, J. F. Frigon, R. G. Bosisio, and K. Wu, “Direct quadrature phase shift keying modulator using six-port technology,” Electron. Lett., vol. 41, no. 21, pp. 1180–1181, Oct. 2005. [10] Y. Zhao, J.-F. Frigon, K. Wu, and R. G. Bosisio, “Multi(six)-port impulse radio for ultra-wideband,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 4, pp. 1707–1712, Jun. 2006. [11] B. Luo and M. Y. W. Chia, “Direct 16 QAM six-port modulator,” Electron. Lett., vol. 44, no. 15, pp. 910–911, Jul. 2008. [12] B. Luo and M. Y. W. Chia, “Performance analysis of serial and parallel six-port modulators,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 9, pp. 2062–2068, Sep. 2008. [13] R. C. Yob, N. Seman, and S. N. A. M. Ghazali, “Error vector magnitude analysis for wideband QPSK and QAM six-port modulator,” in Proc. IEEE Int. RF Microw. Conf. RFM, Seremban, Malaysia, Dec. 2011, pp. 149–153. [14] S. Z. Ibrahim, A. M. Abbosh, and M. A. Antoniades, “Direct quadrature phase shift keying modulation using compact wideband six-port networks,” IET Microw., Antennas Propag., vol. 6, no. 8, pp. 854–861, Jun. 2012. [15] N. S. A. Arshad, W. L. Cheor, S. Z. Ibrahim, and M. S. Razalli, “QPSK modulation using multi-port device,” in Proc. IEEE Symp. Wireless Technol. Appl. (ISWTA), Kota Kinabalu, Malaysia, Sep. /Oct. 2017, pp. 58–63. [16] X. Song et al., “Integrating baseband-over-fiber and six-port direct modulation for high-speed high-frequency wireless communications,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, USA, May 2016, pp. 1–4. [17] B. Zouggari, C. Hannachi, E. Moldovan, and S. O. Tatu, “Millimeter wave six-port QPSK modulators for high data-rate wireless communications,” in Proc. 46th Eur. Microw. Conf. (EuMC), London, U.K., Oct. 2016, pp. 1035–1038. [18] J. Osth, Owais, M. Karlsson, A. Serban, S. Gong, and P. Karlsson, “Direct carrier six-port modulator using a technique to suppress carrier leakage,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 3, pp. 741–747, Mar. 2011. [19] J. Osth, Owais, M. Karlsson, A. Serban, and S. Gong, “Schottky diode as high-speed variable impedance load in six-port modulators,” in Proc. IEEE Int. Conf. Ultra-Wideband (ICUWB), Bologna, Italy, Sep. 2011, pp. 68–71. [20] J. Osth, M. Karlsson, A. Serban, and S. Gong, “Carrier leakage suppression and EVM dependence on phase shifting network in six-port modulator,” in Proc. Int. Conf. Microw. Millim. Wave Technol. (ICMMT), Shenzhen, China, May 2012, pp. 1247–1250. [21] J. Osth, A. Serban, M. Karlsson, and S. Gong, “LO leakage in sixport modulators and demodulators and its suppression techniques,” in IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 2012, pp. 1–3. [22] H. Moazzen, A. Mohammadi, and R. Mirzavand, “Multilevel outphasing system using six-port modulators and Doherty power amplifiers,” Analog Integr. Circuits Signal Process., vol. 90, no. 2, pp. 361–372, Feb. 2017. [23] H. S. Lim, W. K. Kim, J. W. Yu, H. C. Park, W. J. Byun, and M. S. Song, “Compact six-port transceiver for time-division duplex systems,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 5, pp. 394–396, May 2007. [24] W. Ciccognani, M. Ferrari, F. Giannini, and E. Limiti, “A novel broadband MMIC vector modulator for V-band applications,” Int. J. RF Microw. Comput.-Aided Eng., vol. 20, no. 1, pp. 103–113, Jan. 2010. [25] A. Serban, J. Osth, Owais, M. Karlsson, S. Gong, J. Haartsen, and P. Karlsson, “Six-port transceiver for 6–9 GHz ultrawideband systems,” Microw. Opt Technol. Lett., vol. 52, no. 3, pp. 740–746, Mar. 2010. [26] Owais, J. Osth, and S. Gong, “Differential six-port modulator,” in Proc. Int. Conf. Wireless Commun. Signal Process. (WCSP), Nanjing, China, Nov. 2011, p. 4.

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[27] W.-S. Lee, K.-S. Oh, D.-Z. Kim, and J.-W. Yu, “Direct six-port modulator using polyphase networks,” Microw. Opt. Technol. Lett., vol. 53, no. 10, pp. 2321–2324, Oct. 2011. [28] A. Serban, M. Karlsson, J. Östh, Owais, and S. Gong, “Differential circuit technique for six-port modulator and demodulator,” in IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 2012, 3 pp. [29] J. Osth, M. Karlsson, A. Serban, and S. Gong, “M-QAM six-port modulator using only binary baseband data, electrical or optical,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 6, pp. 2506–2513, Jun. 2013. [30] J. Osth, M. Karlsson, A. Serban, and S. Gong, “A comparative study of single-ended vs. differential six-port modulators for wireless communications,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 2, pp. 564–570, Feb. 2015. [31] N.-C. Kuo, J.-L. Kuo, and H. Wang, “Novel MMIC power amplifier linearization utilizing input reflected nonlinearity,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 3, pp. 542–554, Mar. 2012. [32] T. Sharma, R. Darraji, and F. Ghannouchi, “A methodology for implementation of high-efficiency broadband power amplifiers with secondharmonic manipulation,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 63, no. 1, pp. 54–58, Jan. 2016. [33] Y. Zhang et al., “Characterization for multiharmonic intermodulation nonlinearity of RF power amplifiers using a calibrated nonlinear vector network analyzer,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 9, pp. 2912–2923, Sep. 2016. [34] T. Sharma, R. Darraji, F. Ghannouchi, and N. Dawar, “Generalized continuous class-F harmonic tuned power amplifiers,” IEEE Microw. Compon. Lett., vol. 26, no. 3, pp. 213–215, Mar. 2016. [35] J. Kim and K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory,” Electron. Lett., vol. 37, no. 23, pp. 1417–1418, Nov. 2001. [36] F. M. Ghannouchi and O. Hammi, “Behavioral modeling and predistortion,” IEEE Microw. Mag., vol. 10, no. 7, pp. 52–64, Dec. 2009. [37] S. Saied-Bouajina, O. Hammi, M. Jaidane-Saidane, and F. M. Ghannouchi, “Experimental approach for robust identification of radiofrequency power amplifier behavioural models using polynomial structures,” IET Microw. Antennas Propag., vol. 4, no. 11, pp. 1818–1828, Nov. 2010. [38] F. F. Tafuri, C. Guaragnella, M. Fiore, and T. Larsen, “Linearization of RF power amplifiers using an enhanced memory polynomial predistorter,” in Proc. NORCHIP, Copenhagen, Denmark, Nov. 2012, p. 3. [39] J. Xia, A. Islam, H. Huang, and S. Boumaiza, “Envelope memory polynomial reformulation for hardware optimization of analog-RF predistortion,” IEEE Microw. Compon. Lett., vol. 25, no. 6, pp. 415–417, Jun. 2015. [40] F. Mkadem, A. Islam, and S. Boumaiza, “Multi-band complexityreduced generalized-memory-polynomial power-amplifier digital predistortion,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 6, pp. 1763–1774, Jun. 2016.

Weiwei Zhang (GS’16) received the B.Eng. degree in electrical engineering from the Wuhan University of Technology, Wuhan, China, in 2011. He is currently pursuing the Ph.D. degree in electrical engineering at the Beijing University of Posts and Telecommunications, Beijing, China. Since 2015, he has been a Visiting Ph.D. Student with the Intelligent RF Radio Laboratory, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada. His current research interests include microwave power dividers, radio-frequency power amplifiers, and multiport transceivers.

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Abul Hasan (GS’11) received the B.Tech. degree in electronics and communication engineering from IIT Guwahati, Guwahati, India, in 2008, and the M.Sc. degree in electrical and computer engineering from the Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, in 2012, where he is currently pursuing the Ph.D. degree in electrical and computer engineering. He was a Senior Hardware Design Engineer with Geodesic Ltd., Bengaluru, India. His current research interests include multiport techniques and their applications, reconfigurable microwave and RF circuits and systems design, and signal processing for modern communication systems. Fadhel M. Ghannouchi (F’07) has held numerous invited positions with several academic and research institutions in Europe, North America, and Japan. He has provided consulting services to a number of microwave and wireless communications companies. He is currently a Professor and the iCORE/Canada Research Chair with the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, and the Director of the Intelligent RF Radio Laboratory. He has authored or co-authored over 650 publications. He holds ten U.S. patents with five pending patents. His current research interests include microwave instrumentation and measurements, nonlinear modeling of microwave devices and communications systems, the design of power and spectrum efficient microwave amplification systems, and the design of intelligent RF transceivers for wireless and satellite communications. Mohamed Helaoui (S’06–M’09) received the M.Sc. degree in communications and information technology from the École Supérieure des Communications de Tunis, Tunis, Tunisia, in 2003, and the Ph.D. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2008. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary. He has authored or co-authored over 60 publications and has 7 pending patents. His current research interests include digital signal processing, power efficiency enhancement for wireless transmitters, switching mode power amplifiers, and advanced transceiver design for software-defined radio and millimeter-wave applications. Dr. Helaoui is a member of the COMMTTAP Chapter of the IEEE Southern Alberta Section.

Yongle Wu (M’12–SM’15) received the B.Eng. degree in communication engineering and Ph.D. degree in electronic engineering from the Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2006 and 2011, respectively. In 2010, he joined the City University of Hong Kong, Hong Kong, as a Research Assistant. In 2011, he joined BUPT, where he is currently a Full Professor with the School of Electronic Engineering. His current research interests include microwave components and wireless systems design.

Lingxiao Jiao received the bachelor’s degree in electronic science and technology from the Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2014, where he is currently pursuing the Ph.D. degree. In 2014, he joined BUPT, where he is focused on his research. His current research interests include microwave passive components and microwave transceivers.

Yuanan Liu (M’92) received the B.E., M. Eng., and Ph.D. degrees in electrical engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1984, 1989, and 1992, respectively. In 1984, he joined the 26th Institute of Electronic Ministry of China, Langfang, China, where he was involved in the development of the inertia navigating system. In 1992, he held a post-doctoral position with the EMC Laboratory, Beijing University of Posts and Telecommunications (BUPT), Beijing, China. In 1995, he held a second post-doctoral position with the Broadband Mobile Laboratory, Department of System and Computer Engineering, Carleton University, Ottawa, ON, Canada. Since 1997, he has been a Professor with the Wireless Communication Center, College of Telecommunication Engineering, BUPT, Beijing, where he is involved in the development of next-generation cellular systems, wireless LANs, Bluetooth application for data transmission, electromagnetic compatibility design strategies for highspeed digital systems, and electromagnetic interference and expected value of mean square measuring sites with low cost and high performance.

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Direct Error-Searching SPSA-Based Model Extraction for Digital Predistortion of RF Power Amplifiers Noel Kelly , Student Member, IEEE and Anding Zhu , Senior Member, IEEE Abstract— This paper presents a low-complexity architecture to extract model coefficients for digital predistortion of radio frequency power amplifiers. The proposed approach directly updates the model coefficients online using a stochastic optimization algorithm that utilizes random perturbation of the model coefficients to determine the coefficient updating direction and converge toward the optimum solution. This technique avoids resource-intensive matrix operations and the requirement for an offline error model in the conventional model extraction techniques and thus drastically reduces the implementation complexity. The complete model extraction solution has been implemented on a field-programmable gate array, and it is shown that the hardware resource usage is remarkably low. Experimental measurements were conducted on a gallium nitride Doherty amplifier excited by Long Term Evolution signals and the results showed that the proposed technique can achieve linearization performance comparable to that obtained by using the conventional and significantly more complex solutions. Index Terms— Digital predistortion (DPD), linearization, model extraction, power amplifier (PA), stochastic optimization.

I. I NTRODUCTION

D

IGITAL predistortion (DPD) is an advanced linearization technique that is now widely used to compensate for nonlinear behavior of radio frequency (RF) power amplifiers (PAs) in modern wireless communication systems [1], [2]. DPD uses an inverse model of the nonlinear PA to predistort the input signal at digital baseband. To maximize linearity improvement, an accurate behavioral model is required. In recent years, a range of advanced behavioral models for RF PAs have been developed, including modified versions of the Volterra series [3]–[5] and, more recently, the decomposed vector rotation (DVR)-based model which uses the absolute value operator as the basis function [6]. To extract the coefficient values for these models, least squares (LS)-based algorithms are typically used [7]–[9]. The LS algorithm offers high accuracy and fast convergence but comes with a high implementation cost in terms of hardware

Manuscript received May 10, 2017; revised July 10, 2017; accepted August 3, 2017. Date of publication October 16, 2017; date of current version March 5, 2018. This work was supported in part by the Science Foundation Ireland and in part by the European Regional Development Fund under Grant 13/RC/2077 and Grant 12/IA/1267. (Corresponding author: Noel Kelly.) The authors are with the School of Electrical and Electronic Engineering, University College Dublin, 4 Dublin, Ireland (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2748128

resources as it requires complex matrix multiplication and inversion operations [10]. High implementation complexity is particularly undesirable for applying DPD in small-cell base stations that are expected to form a large part of future 5G networks. These stations operate at much lower power levels than those in conventional larger cells. The power efficiency and implementation cost of all components in the transmitter chain, including DPD, must be carefully managed [11], [12]. To reduce the computational complexity, iterative coefficient extraction techniques can be considered. In particular, the recursive least-squares (RLS) algorithm has been employed in DPD model extraction [13], [14]. RLS avoids large matrix inversion, but maintaining an accurate approximation of the Hessian matrix still requires significant complexity, particularly for higher order models. In [15] and [16], the authors proposed a model adaption technique based on the least mean squares (LMS) algorithm where large matrix calculations are avoided and thus implementation complexity is greatly reduced. However, because it uses the first-order approximation, LMS is very sensitive to the adaption step size and it typically struggles to achieve the desired model accuracy [17]. In [18], a stochastic optimization-based DPD coefficient calculation technique was proposed as a low-complexity alternative to the LS solution. It is derived from the simultaneous perturbation stochastic approximation (SPSA) algorithm that uses measurements of the loss function with a random perturbation on the model coefficients to determine the coefficient updating direction and converge toward the optimum solution without involving resource-intensive matrix operations [19], [20]. It is shown in [18] that, after a sufficient number of iterations, the technique can achieve accuracy comparable to the existing LS solutions with over 98% reduction in computational complexity. However, to enable quadratic interpolation, new error model outputs must be calculated at each SPSA iteration. If a large number of training samples are used, the calculation of error model outputs still requires a large number of operations, leading to significant resource usage and cost. This paper presents an alternative training architecture that removes the requirement for an additional error model in the coefficient extraction procedure. The proposed technique applies the SPSA algorithm directly on the DPD model to find the optimum coefficients. Experimental results show that the proposed approach can achieve comparable linearization performance to existing solutions but offers a further drastic reduction in hardware resource usage compared with the existing technique in [18].

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KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

Fig. 1.

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DPD with direct learning model extraction.

This paper is organized as follows. In Section II, the existing direct learning methods using both LS and the SPSA algorithm are discussed. The direct learning model extraction solution proposed in this paper is detailed in Section III. Section IV outlines the implementation complexity of the proposed technique and quantifies the improvement by comparing it against the existing SPSA-based DPD. Finally, Section V reports experimental results with a conclusion in Section VI. II. E XISTING D IRECT L EARNING M ODEL E XTRACTION The principle of DPD is that a digital block is inserted into the transmitter chain to preprocess the input signal before it enters the RF PA. Two primary architectures are commonly used for model extraction: indirect learning (IDL) and direct learning [2], [9], [21]. The IDL architecture estimates the postinverse of the PA first and then copies the coefficients to the preinverse DPD block. The direct learning architecture is usually used in a closed-loop system and it directly compares the PA output with the original input. A. LS-Based Direct Learning Fig. 1 shows the block diagram of a DPD system with direct learning. Many behavioral models can be used for constructing the DPD function. In this paper, we use the DVR-based model [6]. The predistorted signal u(n) ˜ is given by u(n) ˜ =

M 

a˜ i x(n ˜ − i)

+

M S   s=1 i=0 M S  

  ˜ − i )| − βs e j θ(n−i) c˜s,i,1 |x(n

C = [a˜ 1 , a˜ 2 , · · · , c˜s,1,1 , · · · ].

(2)

The direct learning model extraction architecture iteratively adjusts the coefficients in C to minimize the error between y and x. As shown in Fig. 1, this means the DPD model is located inside the training loop. The update equation is given by Ch+1 = Ch − λ · Ch

(3)

where λ is a scalar adaption factor and h is the iteration index. Ch is the coefficient updating vector modeling the error component in the DPD coefficient vector. Provided the error is sufficiently small, the error in the DPD output signal, u error,h , can be approximated by the error in the PA output uerror,h = G −1 (yh − x) ≈ yh − x

(4)

where G −1 (·) is the inverse transfer function of the PA. The coefficient update vector Ch can then be estimated using LS Ch = (X H X)−1 X H (uerror,h )

(5)

B. SPSA-Based Direct Learning

  c˜s,i,21 |x(n ˜ ˜ − i )| − βs e j θ(n−i) |x(n)|

s=1 i=0

+...

samples of the signals x(n), ˜ u(n), ˜ and y˜ (n) in Fig. 1 using the notation x, u, and y, respectively. For convenience, we also group the model coefficients in (1) into a single coefficient vector

where X is the DPD model regression matrix generated using the DPD input signal x [22].

i=0

+

Fig. 2. SPSA-based direct learning DPD [18]. (Note the symbol “⊗” represents the Kronecker product.)

(1)

where x(n) ˜ is the baseband input signal. The constants M and S are the memory length and the number of thresholds, respectively. The operator | · | denotes the absolute value operation and θ (n) is the phase of the signal x(n). ˜ It is common to consider DPD processing to take place in blocks of N samples. In this paper, we represent vectors containing

The LS operation in (5) involves large matrix operations, which are hardware demanding and time consuming. To reduce complexity, in [18], we proposed to replace LS with the SPSA algorithm. SPSA is a stochastic optimization algorithm that iteratively measures a loss function with a random perturbation on the model coefficients to determine the coefficient updating direction and finally find the optimum solution. The coefficient perturbation process only requires a simple addition and subtraction operation, and all coefficients are randomly perturbed together, which leads to substantial savings in hardware resource usage in model extraction.

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To replace LS with SPSA to calculate Ch in (5), we first need to construct an error model with output emod given by emod = XCerr .

(6)

The goal of the SPSA process is to find the optimum error coefficient vector Cerr to enable emod approach uerror , so that Cerr can then be used to replace Ch in (3) to update the DPD coefficients in the next direct learning iteration. As mentioned above, SPSA is an iterative search algorithm and thus, as shown in Fig. 2, multiple internal iterations are required to find the optimum Cerr coefficients each time. The iteration procedure is outlined below. Note that we use the index k to denote internal iterations as opposed to the index h in (3). The iteration begins with perturbing the current estimate of the error model coefficient vector Cerr,k with a random perturbation vector, k , weighted by a scalar, ck , to generate two additional coefficient vectors C+ err,k = Cerr,k + ck k

C− err,k = Cerr,k − ck k .

(7)

The perturbation vector k is given by k = [k,1 , k,2 , · · · , k,K ].

(8)

where K is the number of terms in the model and each entry k,i is either +1 or −1 with equal probability. Using the perturbed coefficients in (7) and the current coefficient set Cerr,k , three error model outputs are calculated emod,k = XCerr,k + emod,k = XC+ err,k

− emod,k = XC− err,k .

(9)

By comparing with the desired error vector uerror , three normalized mean square error (NMSE) loss function measure− ments, L(Cerr,k ), L(C+ err,k ), and L(Cerr,k ) can be obtained. As discussed in [18], since the NMSE is quadratically related to the model coefficients, the next updated coefficient estimate, Cerr,k+1 , can be found by moving directly to the minimum point of the quadratic curve formed by the three loss function measurements. Specifically, the new error coefficient vector is calculated by Cerr,k+1 = Cerr,k − ck k μk

(10)

where the perturbation update weighting, μk is given by1   −   L C+ err,k − L Cerr,k     (11) μk =   +  2 L Cerr,k + L C− err,k − 2L Cerr,k As shown in Fig. 3, as the updating process iterates, the NMSE is reduced in each iteration. The training finishes when the desired accuracy is reached. This method avoids the gradient calculations and resulting resource-intensive matrix operations required by LS. 1 Due to a typing error, the scaling factor of 2 was missing in the

denominator part of the quadratic SPSA coefficients updating equation given in [18]. It is included in (11) now.

Fig. 3.

Evolution of the quadratic SPSA interpolation.

As shown in [18], this approach achieves a reduction in periteration computational complexity of over 98%, while maintaining comparable performance to conventional LS. However, in terms of implementation complexity, the introduction of a secondary error model is not desirable. At each SPSA iteration, new error model outputs must be calculated according to the expressions in (9). Note that the matrix Xh has dimensions N × K , where N is the number of training samples and K is the number of terms in the DPD model. In a practical system, this calculation requires a large number of operations, leading to significant resource usage and cost. III. D IRECT E RROR -S EARCHING SPSA-BASED M ODEL E XTRACTION To further reduce the implementation complexity of the technique in [18], this paper proposes a novel extraction method where the SPSA algorithm is applied directly to the DPD model rather than to the error model. A. Error Analysis In a DPD system, shown in Fig. 1, the goal of the model extraction process is to find a set of ideal coefficients that can generate the ideal output uideal that enters the PA to generate the perfect output, yideal that is equal to the original input x. Clearly, uideal is not available before we find the ideal coefficients, but it can be expressed as uideal = uh − uerror,h

(12)

where uh is the existing DPD output that can be obtained by uh = XCh

(13)

where Ch is the model coefficients vector. As in (4), for a given set of DPD coefficients, the error signal uerror,h at the DPD output can be approximated by the error measured at the PA output, if the error is relatively small [22], [23]. Substituting (4) into (12) gives an approximate value for the ideal DPD output uideal ≈ uh − (yh − x).

(14)

If we know uideal , the optimization task now is to find the ideal DPD coefficients, Cideal , that minimize the error between the

KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

Fig. 5. Fig. 4.

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Quadratic fitting.

Single SPSA iteration of the proposed algorithm.

DPD model output, uh , and its ideal value, uideal   Cideal = arg minCh |uh − uideal |2 .

(15)

Considering the quadratic feature of SPSA, this optimization problem can be solved by using SPSA directly, as illustrated in Fig. 4, where the coefficients Ch are on the horizontal axis, while the magnitude square of the output error signal |u error,h |2 is on the vertical axis. Because both uh and uideal are linearly related to the model coefficients, uerror,h is also linearly related to Ch . Therefore, |u error,h |2 is quadratic in relation to Ch . It follows that the next error and the corresponding coefficients can be found by simply forming a quadratic curve using SPSA. As shown in Fig. 4, assuming that Ch corresponds to the existing error |u error,h |2 , if we perturb the coefficients to − 2 2 generate two other errors |u + error,h | , |u error,h | , we can form a quadratic curve. The minimum point of the curve is the next error value |u error,h+1 |2 , and thus, the new corresponding coefficients Ch+1 can be found. The difference compared with the existing approaches described in Section II is shown as Existing: Ch ⇒ uerror,h ⇒ Ch ⇒ Ch+1 Proposed : Ch ⇒ uerror,h ⇒ uerror,h+1 ⇒ Ch+1

(16)

where in the existing approaches, the next coefficients vector is updated using an error coefficients vector generated from the existing error, while in the proposed approach, we find the next DPD error from the SPSA quadratic curve fitting and thus find the next optimum coefficient vector directly. This approach is much simpler in terms of computational complexity and hardware implementation compared with the existing approaches, discussed as follows. B. SPSA Training To apply SPSA, we first define a loss function. In this paper, we propose to use a loss function that is given as the residual sum of squares (RSS) between the DPD output u and the ideal DPD output uideal for a given set of measurements RSS(uideal , u) =

N  n=1

|u(n) ˜ − u˜ ideal (n)|2

(17)

where N is the total number of training samples. It is similar to the standard NMSE. The difference is that the division operation in the NMSE definition is no longer required here as uideal is constant across all three measurements during a single SPSA iteration. As discussed earlier, to perform quadratic SPSA, three loss function measurements are required. One measurement is conducted by using the existing coefficients Ch , while the other two measurements are obtained by applying a random perturbation vector h , weighted by a scalar ch , to the existing coefficients Ch to generate two additional DPD output signals. Intuitively, we would think that we have to feed the coefficients through the DPD block three times to generate three DPD outputs, which will complicate the process. In fact, by taking advantage of the quadratic property of SPSA, it is possible to obtain the other two outputs with simple addition and subtraction operations on the existing output, explained as follows. First of all, note that the two additional perturbed DPD outputs are used to form the quadratic curve and find the next DPD error only. They will not enter the PA to produce new final outputs. The perturbation error levels therefore do not affect the real-time system operation. Second, because we have ensured a quadratic relationship between the coefficients and loss function, with the same input signal, all possible loss function measurements lie on the same quadratic curve, no matter what weighting factor ch is used. As shown in Fig. 5, two solid square points are generated from ch = 1, while two solid round dot points are generated with ch = 0.15. All four points lie on the same quadratic curve. This means that no matter what weighting factor is used, the next minimum point can always be found after three measurements. Although the perturbation errors are bigger in the case of ch = 1 than those with ch = 0.15, these errors do not affect the real system operation since the perturbed coefficients will not be used directly in the real-time DPD operation. As a result, to reduce computational complexity, here we choose ch = 1 and the perturbed model outputs, u+ and u− are given by u+ = X(Ch + h ) u− = X(Ch − h ).

(18)

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Loss function measurement implementation.

Fig. 7.

Proposed direct error-searching SPSA model extraction.

Fig. 8.

Proposed model training routine.

Expanding (18) gives u+ = X(Ch + h ) = uh + (Xh ) u− = X(Ch − h ) = uh − (Xh )

(19)

meaning the coefficient perturbation can be expressed as an additional signal, uh , to be added to and subtracted from the original DPD output u+ = uh + uh u− = uh − uh

(20)

uh = Xh .

(21)

where

Since the elements in h are either +1 or −1, uh in (21) can be calculated with simple addition and subtraction operations between each column of the model regression matrix, X defined in Section II-A ⎡ ⎤ ⎡ ⎤ −1 X 1,1 X 1,2 X 1,3 · · · ⎢ ⎥ ⎢ X 2,1 X 2,2 X 2,3 · · ·⎥ ⎢+1⎥ ⎣ ⎦ ⎢−1⎥ .. .. .. .. ⎣ ⎦ .. . . . . . ⎤ ⎡ −X 1,1 + X 1,2 − X 1,3 + · · · ⎥ ⎢ = ⎣−X 2,1 + X 2,2 − X 2,3 + · · ·⎦ (22) .. . After obtaining uh and considering (14) and (17), three loss function measurements can be conducted L (Ch ) =

N 

2 | y˜h (n) − x(n)| ˜

n=1

  L C+ h =

N 

  L C− h =

N 

| y˜h (n) − x(n) ˜ + u˜ h (n)|2

n=1

| y˜h (n) − x(n) ˜ − u˜ h (n)|2

(23)

n=1

as shown in Fig. 6. The results from (23) along with the three coefficients sets can be used to form a quadratic curve for each coefficient and then new coefficients can be found by using the same equations in (10) and (11). The updated DPD coefficients produce a closer match to uideal at the DPD model output. This operation significantly simplifies the updating process and reduces system cost. More details on the digital implementation will be given in Section IV.

The fact that the measurements in (23) are performed at the DPD output is important for two reasons. First, the desired quadratic interpolation SPSA algorithm can be directly applied. Second, the three loss function measurements can be conducted at the same time and there is no need to feed the two perturbed DPD output signals through the PA to measure the loss function, which means that the model training process does not disrupt the real-time DPD operation. The block diagram of the proposed full system is shown in Fig. 7. C. Complete Model Extraction Procedure Despite the use of the term “ideal” in the notation, the approximation in (4) limits the accuracy of the calculated Cideal coefficients. The signal uideal is not the “truly” ideal DPD output but rather an estimated ideal output based on the approximation in (4). Thus, even if we could train the DPD coefficients to exactly fit uideal , the error at the PA output would not be completely removed. The traditional direct learning architecture faces an identical problem. Furthermore, in the above training, we assume uideal is generated from a fixed set of input signal samples x. In real operation, the input signal x is randomly generated over time and a different x, e.g., xh corresponds to a new uideal,h . To perform accurate coefficient extraction in this environment, multiple iterations of the coefficient extraction process are required. The complete model training flow is depicted in Fig. 8. The training procedure can be described as follows. 1) Set iteration index, h = 1, and choose initial DPD coefficient set C1 .

KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

2) Measure the PA output signal, yh . 3) Calculate the approximate DPD error signal, uerror,h according to (4). 4) Calculate the ideal DPD signal, uideal according to (14). 5) Measure RSS (uideal , uh ), RSS uideal , uh+ , and  RSS uideal , uh− . 6) Calculate Ch+1 using the SPSA update equations in (10) and (11) to minimize error between DPD output uh and uideal . 7) Generate new DPD output using Ch+1 and pass to the PA. If linearization criterion is satisfied, finish training, if not, update h = h + 1 and return to step 2. This iterative training process shares similarities with the conventional direct learning procedure. At each iteration, the DPD coefficients are updated to approach an estimate of the ideal predistorted signal. Provided the approximation in (14) holds, the error at the PA output is reduced after each DPD coefficient update. It follows that the accuracy of the approximation in (4), on which (14) is based, also improves with each update and the DPD coefficients are trained to approach a more accurate estimate of the ideal output at each iteration. D. Comparison With the Existing Approaches Although an iterative coefficient updating process is employed, the proposed technique in this paper is fundamentally different from the conventional iterative approaches. In the existing systems, LMS may be employed but its performance is poor because LMS is a first-order approximation algorithm that converges very slowly and it is sensitive to the adaptation size. The second-order approximation approaches, such as LS and RLS, can achieve high accuracy but come with high implementation complexity. The proposed approach is also different from the conventional SPSA where the gradient is approximated by the first-order line fitting. Such linear approximations are highly sensitive to the choice of weighting factor and can often struggle to converge [19]. By exploiting the quadratic relationship that exists between the loss function and the DPD coefficients, the optimum minimum point can be found directly using quadratic curve fitting instead of gradient approximation. This approach is equivalent to the second-order approximation that significantly speeds up the convergence and guarantees the optimum point can be found at each iteration. Furthermore, due to the quadratic relationship, the perturbation step size is no longer relevant because all the cost function measurement points will fall on the same curve which leads that we could use any perturbation step size. In this paper, we directly use +/ − 1, which enables the cost function to be directly obtained by adding and subtracting the basis waveforms that have already been generated by the DPD model, dramatically simplifying the hardware implementation. While the optimization process is changed to the secondorder approximation, the proposed approach still keeps the core feature of SPSA, namely, the optimum solution is found using loss function measurements with simultaneous random perturbation on model coefficients. It enables the algorithm to achieve comparable accuracy to LS solutions but with a

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TABLE I SD-SPSA P ER -I TERATION C OMPLEXITY [18]

very low implementation cost. We call the proposed approach “direct error-searching” SPSA-based model extraction. IV. H ARDWARE I MPLEMENTATION AND C OMPLEXITY C OMPARISON To quantify the improvement offered by the direct errorsearching SPSA approach outlined above, we compare it with the generic SPSA-based DPD extraction method proposed in [18], which we refer to from here on as SD-SPSA. It is worth noting that the SD-SPSA method has already achieved a 98% reduction compared with conventional LS algorithms, as discussed in detail in [18]. To avoid replication, in this paper, we only discuss further reduction from the SD-SPSA solution, without comparison with LS. A. Operations per Iteration Both the SD-SPSA and the direct error-searching SPSA use multiple iterations, in this section, we compare the complexity per iteration. The SD-SPSA solution directly replaces LS with SPSA in the closed-loop direct learning architecture. In this case, the SPSA algorithm calculates the error coefficients, Cerr , using an error model in the same format as the LS estimation technique would be applied. The solution in [18] employs a novel steep descent SPSA algorithm to increase the training speed and results showed that the linearization performance is comparable with the existing LS solutions. However, as discussed in Section II-B, additional processing associated with the error model substantially increases resource usage. Table I reports the operations that account for the majority of the real multiplication and addition operations used in a single iteration of the algorithm in [18]. Generating the error model output accounts for the vast majority of the complexity. Taking a typical example of a DPD model with 50 terms (K = 50) and using 8192 samples to calculate the NMSE on each iteration (N = 8192), running the error model accounts for over 98% of the total real multiplications and over 99% of the total real addition operations performed each iteration. The solution proposed in this paper removes the need for an error model. It finds the next coefficients directly by perturbing

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TABLE II P ROPOSED A PPROACH P ER -I TERATION C OMPLEXITY

Fig. 9.

Proposed hardware implementation accuracy.

TABLE III

the DPD model. As discussed in Section III-B, the real and imaginary components of each term in h are limited to values of +/ − 1. In this scenario, the uh vector can be calculated using only simple addition and subtraction operations across the columns of the matrix X. This amounts to a substantial reduction in the number of operations required per iteration and also allows the SPSA update equation to be evaluated without the need for costly multiplication operations. Table II reports the number of real multiplication and addition operations required per iteration for the proposed approach. Removing the need to run an offline error model each iteration substantially reduces the number of multiplication operations required. To quantify the improvement, resource usage for a practical training scenario with K =50 and N =8192 is shown in brackets. Referring back to Table I, in this scenario, the SD-SPSA technique in [18] requires approximately 2.5 × 106 real multiplications and 5.7 × 106 real additions. By comparison, as shown in Table II, the proposed error-matching SPSA uses approximately 5×104 real multiplications and 1.7 × 106 real additions. It is also important to note that, in addition to the complexity reported in Table I, the technique in [18] requires the DPD matrix X and the captured PA output y to remain constant throughout the offline training segment. This leads to a complexity tradeoff in the design. On the one hand, as proposed in [18], the matrix X can be stored after it has been generated by the DPD model and reused for each SPSA iteration, but this requires a large block of memory resources. On the other hand, the system could store only the input signal x and implement a secondary offline model to generate X each iteration, reducing the memory requirements but significantly increasing the computational complexity. B. Hardware Implementation Complexity The proposed extraction solution was synthesized for implementation in a field-programmable gate array (FPGA). The designed system performs all of the computation necessary to update the model coefficients each iteration: it takes the signals, x, X, and y as inputs and processes them to generate the SPSA update vector.

H ARDWARE I MPLEMENTATION D ETAILS

The complex samples are represented using 32-b precision, 16 b each for the real and imaginary components. A hardware simulation was performed to confirm the design feasibility. The simulation used digital baseband samples and a full precision model of the DPD-PA transmit chain. The DPD model is a DVR-based function with S = 8 and M = 3, and the PA is modeled using a dynamic deviation reduction-based Volterra series with nonlinear terms up to the ninth order and memory length 3 [8]. The PA model is based on data measured from a gallium nitride (GaN) Doherty PA. A 20-MHz Long Term Evolution (LTE) signal serves as the system input. The primary objective of the simulation is to confirm that the 16-b implementation provides sufficient precision that the algorithm convergence is not affected. Fig. 9 shows the NMSE measured after each iteration between the DPD output signal generated using the 16-b FPGA coefficients and the output signal generated using full precision coefficients. The NMSE remains below −70 dB throughout the 40 000 iteration training period, confirming there is no significant degradation due to rounding error. Fig. 9 also reports the NMSE measured between the system input and output for the bit-accurate model extraction scenario. The system NMSE reaches the full precision performance of approximately −43 dB, confirming the implemented design achieves sufficient precision. Table III reports digital hardware resource usage for the implemented system. To demonstrate its simplicity, the complete extraction solution is implemented in lookup tables

KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

(LUTs) and registers only—no specialized DSP units are used. The system loads the DPD matrix columns serially, meaning that the hardware usage reported in Table III is independent of the DPD model length. Serial loading allows a single accumulator to calculate uh . This reduces hardware usage but requires K clock cycles per input sample, where K is the number of terms in the DPD model and thus columns in the DPD matrix. Alternatively, if maximum throughput is required, more than one column may be loaded simultaneously; full parallel loading corresponds to the greatest hardware resource usage but allows a new sample to be processed on every clock cycle. This is one of a number of design tradeoffs that can be made in implementing the proposed system. By requiring only three loss function measurements to perform a coefficient update, the flexibility of the SPSA algorithm permits a large number of possible implementation strategies, tuned to different criteria such as minimum resource usage, maximum data throughput, or minimum power consumption. It is expected that, compared with the proof-of-concept solution reported in Table III, even lower resource usage can be achieved in the future systems by leveraging known application scenarios and employing more advanced hardware implementation tools. Table III also includes power consumption figures for each of the main blocks of the proposed system. The power consumption measurements were obtained using the post implementation power analysis tool in the Xilinx Vivado integrated design environment software. It should be stressed that these figures are reported only to provide the reader with an approximate breakdown of the power consumption in the circuit. In a real implementation, power consumption of the real circuit is highly dependent on the application scenario, e.g., signal bandwidth/sampling rate, number of coefficients used, clock rate, digital circuit chip types (e.g., FPGA part number), and implementation strategy. As a result, the power consumption can vary largely in different cases. Nonetheless, it is interesting to note that the main SPSA operation, i.e., the model coefficient update, only requires an estimated 30 mW to operate. This emphasizes the very low implementation cost of the SPSA algorithm. The majority of the power consumption is due to the loss function calculation. As discussed later in Section VI, the loss function measurement may be implemented in a highly efficient way in the analog domain in the future that may provide the potential for further power reduction. To quantify the reduction in resource usage, we compare the proposed technique with the offline SD-SPSA solution in [18]. In fact, the two designs share much of their infrastructure. We first simplify the SD-SPSA method by swapping NMSE with an RSS loss function and also ignoring the added complexity of the offline steep-descent calculation. The reduced complexity gives the SD-SPSA technique an advantage in the comparisons but allows a clearer comparison between the key features of the systems, namely, generating the loss function measurement signals. For a fair comparison, we use an implementation of the SD-SPSA algorithm in [18], in which the loss function is measured three times each iteration. In terms of computational complexity, the primary advantage of the

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TABLE IV E RROR S IGNAL G ENERATION C OMPLEXITY

TABLE V H ARDWARE R ESOURCE U SAGE C OMPARISON

Fig. 10.

Experimental test setup.

proposed SPSA technique is in generating the RSS input signals, as shown in Fig. 6. Table IV compares guideline resource usage between the SD-SPSA solution and the error-searching SPSA technique. It can be seen that the complex multiplications required to run the error model consume far more resources than the simple addition and subtraction operations of the error-searching technique reported in Table III. Note that the resource usage in Table IV is for an efficient serial-loading implementation of the SD-SPSA algorithm, where only three complex multiplications are required. For higher throughput scenarios, complexity will increase with the number of parallel operations. Table V compares the overall resource usage between the two architectures. In addition to the reduced computational resource requirements (i.e., raw LUTs and flip flops), the proposed technique no longer requires large blocks of signal samples to be stored. In [18], the algorithm requires the full Xh matrix and the measured error vector to be stored during the SPSA training run. Assuming 32-b accuracy for each complex value, for a test scenario with K = 50 and N = 8192, this amounts to a storage requirement of 32 × ((N × K ) + N) = 13 369kb. As shown in Table V, the proposed technique requires no additional memory resources, substantially reducing implementation complexity. V. E XPERIMENTAL R ESULTS A full RF test bench, as shown in Fig. 10, was set up to evaluate the performance of the proposed architecture in a

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TABLE VI 20-MHz LTE M EASUREMENT R ESULTS

Fig. 11.

Measured NMSE over 20 000 iterations with 20-MHz LTE signal.

real DPD training scenario. To evaluate the online coefficient updating process in hardware, an automated test routine was developed to run multiple iterations of the algorithm. The test bench is centered on a master PC which is responsible for the signal generation and control of the measurement equipment to capture the PA input and output signals. DPD coefficients are calculated in the PC according to the proposed SPSA algorithm by using the captured error signal. At the beginning of each iteration, the predistorted signal is loaded onto a Rohde and Schwarz SMW200A vector signal generator where it is up-converted to RF before transmission to the PA. A linear driver boosts the signal power before it enters the DUT, a high-efficiency 20-W GaN Doherty PA operated with an average output power of 36 dBm. At the PA output, an attenuator reduces the signal power before a Rohde and Schwarz FSW signal and spectrum analyzer captures the signal for demodulation, sampling and upload to the PC. This process repeats automatically and the DPD coefficients are updated each iteration until a desired linearization target (e.g., maximum number of iterations or specified NMSE threshold) is reached. Accounting for settling time, instrument setup, and data transfer, each iteration lasts approximately 1 s. At the PC, the DVR-based model in (1) is used to generate the predistorted signal. The model is set up with S = 8 and M = 3. The sampling frequency for the signal generator and spectrum analyzer is 184.32 MHz. A block of 15 000 signal samples are loaded into the signal generator and captured at the signal analyzer each iteration. To recreate a realistic training scenario the input signal (x in Fig. 7) is varied for each new SPSA iteration by selecting a different set of 15 000 contiguous samples from a large stored data set. A. 20-MHz Single-Carrier LTE Signal The performance was first tested using a 20-MHz singlecarrier LTE signal with 6.5 dB peak to average power ratio. The carrier frequency was 1.84 GHz and the average input signal power to the PA was 20.5 dBm. Fig. 11 reports convergence performance in terms of NMSE measured between the system input and output signals. As mentioned above, the input signal is varied on each iteration to recreate a realistic training scenario, this causes the measured NMSE to fluctuate around a decreasing mean value as the algorithm converges.

Fig. 12.

Measured PA output spectra for a 20-MHz LTE signal.

A 500-point moving average for the measured NMSE is included in Fig. 11 to illustrate the overall convergence trend. The reference NMSE measurement taken when the predistortion coefficients are extracted using a standard LS/IDL method is also reported. The convergence curve follows a familiar pattern for SPSA with an initial fast training period that rapidly slows down as the coefficients approach their optimum values. Table VI compares the performance in terms of time domain NMSE and frequency domain adjacent channel power ratio (ACPR) with the conventional LS/IDL method. After 20 000 iterations, the NMSE performance of the proposed method is within 0.1 dB of the conventional LS/IDL reference case, and the difference between the two techniques in terms of ACPR is less than 2 dB. Fig. 12 shows the measured PA output spectra with and without DPD applied. After 20 000 iterations, the linearization performance is close to that of the conventional LS/IDL approach. Fig. 13 shows the AM/AM and AM/PM characteristics before and after linearization using the SPSAcalculated coefficients. Compared with LS, the proposed approach requires a large number of iterations, but as discussed earlier, the computational complexity at each iteration is very low and, thus, the overall operation can be very fast. In a practical implementation, the training time depends on the number of the sampling points used and the sampling rate of the signal at the DPD output. Assuming a sampling rate of 400 MS/s, and capturing 8192 samples per iteration, a 20 000 iteration training run would take approximately 400 ms to complete.

KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

Fig. 13. AM/AM and AM/PM plots for a 20-MHz LTE signal with and without DPD.

Fig. 14.

Measured SPSA convergence for a 40-MHz LTE signal. TABLE VII 40-MHz LTE M EASUREMENT R ESULTS

Fig. 15.

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Measured PA output spectra for a 40-MHz LTE signal.

Fig. 16. AM/AM and AM/PM plots for a 40-MHz LTE signal with and without DPD.

in out-of-band spectral regrowth is comparable between the SPSA and LS/IDL techniques. Finally, the AM/AM and AM/PM curves in Fig. 16 show successful linearization using the SPSA-calculated DPD coefficients. VI. C ONCLUSION

B. 40-MHz Dual-Carrier LTE Signal The proposed technique was also evaluated using a 40-MHz dual-carrier LTE signal with 9.5 dB peak to average power ratio. The carrier frequency was again 1.84 GHz and the average signal power at the PA input was 20.5 dBm. Fig. 14 shows the algorithm convergence over the course of 40 000 iterations. The increased number of iterations illustrates that, although the convergence speed slows as the iteration number increases, the NMSE continues to improve. With 10 000 iterations, the SPSA test case NMSE is approximately 4 dB worse than the LS/IDL reference, however, after 40 000 iterations, the gap between the two is reduced to 1.8 dB. Table VII reports the NMSE and ACPR measurements for the test scenario. In terms of ACPR, after 40 000 iterations, the measured values for the SPSA DPD are within 2 dB of the LS/IDL reference. Fig. 15 shows the linearized output spectrum where the reduction

A low-complexity DPD model extraction technique has been presented. The proposed solution integrates the SPSA algorithm into the direct learning architecture and uses a modified iteration technique for extracting DPD coefficients. Measurement results indicated that the proposed technique can achieve comparable linearization performance to the existing LS-based solutions but with considerably lower implementation cost. Because the algorithm is based on stochastic search, multiple iterations would be required to find the final optimum solution. One may argue that the total computational complexity of the proposed approach, i.e., computation per iteration × number of iterations, may be comparable with or even higher than what LS requires, since LS can converge within a very few iterations while SPSA requires tens of thousands iterations. When making this comparison, a few points should be considered. First of all, it is worth mentioning that a large number of iterations for SPSA training are only required at the system startup. When the DPD system is running in real time, a much smaller number of iterations are typically

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required to keep the performance in the acceptable range. The complexity of real time maintenance is therefore very low. This is in contrast to LS, where the full operation must be conducted at each update cycle. If, therefore, considering a life-time operation, the computational complexity and power consumption of the SPSA approach would be much lower compared with the LS approach. Second, although computational operation is an important concern, there are many other factors that need to be taken into account when implementing a DPD system, including component count, silicon area, and overall system cost, as highlighted in [12]. The LS algorithm offers high accuracy and fast convergence but comes with a high implementation cost in terms of hardware resources as it requires complex matrix multiplication and inversion operations. These operations require special DSP circuits, e.g., dedicated microprocessors, to implement, which can occupy a large silicon area and be costly. High implementation cost is particularly undesirable in 5G smallcell base stations since these small cell stations operate at much lower power levels and the overall cost of the system shall be very low compared with those in conventional larger cells. The cost of each component in the transmitter chain must be carefully managed. Including a complex LS engine in the transmitter for DPD model extraction would not be a favorable solution. Furthermore, in the future wireless systems, in particular small cells, the system will become much more integrated. Including a large silicon area in the transceiver would not be desirable. By employing the iterative approaches, e.g., the proposed SPSA, the required silicon area and cost are fractional compared with that required by LS, which makes them far more attractive. In addition, we shall point out that the proposed SPSAbased technique in this paper is substantially different from the conventional approaches. In the existing algorithms, such as RLS, the computational complexity is heavily dependent on the number of coefficients and the model structure used. In the proposed approach, all the model coefficients are perturbed at the same time and they are extracted based solely on measurements of the loss function instead of gradient calculation. The model extraction is thus independent of the number of coefficients and it does not require knowledge of the model structure or nonlinear term construction. It makes model extraction much more flexible and the new coefficient set can be generated with very few operations. Currently, the loss function measurement, i.e., RSS calculation, consumes the majority of power. It is envisioned that, in the future systems, the loss function may be implemented in a highly power efficient manner in the analog domain that can enable the total power consumption to be further reduced. This is a unique ability offered by the proposed approach. In conclusion, although the total number of operations may be comparable with the conventional LS when all iterations are considered, the SPSA-based approach has many unique advantages. These advantages make the proposed technique as an attractive solution for DPD systems in the future 5G small-cell networks, where energy efficiency and cost-effective implementation are expected to become critical requirements.

R EFERENCES [1] J. G. Wood, Behavioral Modeling and Linearization of RF Power Amplifiers. Norwood, MA, USA: Artech House, 2014. [2] R. N. Braithwaite, “General principles and design overview of digital predistortion,” in Digital Front-End in Wireless Communications and Broadcasting, F.-L. Lou, Ed. Cambridge, U.K.: Cambridge Univ. Press, 2011, ch. 6, pp. 143–191. [3] F. M. Ghannouchi and O. Hammi, “Behavioral modeling and predistortion,” IEEE Microw. Mag., vol. 10, no. 7, pp. 52–64, Dec. 2009. [4] D. R. Morgan, Z. Ma, J. Kim, M. G. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3852–3860, Oct. 2006. [5] A. Zhu, J. C. Pedro, and T. J. Brazil, “Dynamic deviation reductionbased Volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4323–4332, Dec. 2006. [6] A. Zhu, “Decomposed vector rotation-based behavioral modeling for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 2, pp. 737–744, Feb. 2015. [7] L. Ding et al., “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, no. 1, pp. 159–165, Jan. 2004. [8] A. Zhu, P. J. Draxler, J. J. Yan, T. J. Brazil, D. F. Kimball, and P. M. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based Volterra series,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 7, pp. 1524–1534, Jul. 2008. [9] C. Eun and E. J. Powers, “A new Volterra predistorter based on the indirect learning architecture,” IEEE Trans. Signal Process., vol. 45, no. 1, pp. 223–227, Jan. 1997. [10] L. Guan and A. Zhu, “Optimized low-complexity implementation of least squares based model extraction for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 3, pp. 594–603, Mar. 2012. [11] R. Q. Hu and Y. Qian, “An energy efficient and spectrum efficient wireless heterogeneous network framework for 5G systems,” IEEE Commun. Mag., vol. 52, no. 5, pp. 94–101, May 2014. [12] J. Wood, “System-level design considerations for digital pre-distortion of wireless base station transmitters,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 5, pp. 1880–1890, May 2017. [13] N. Zheng, Y. Chen, X. Wu, and J. Shi, “Digital predistortion based on QRD-RLS algorithm and its implementation using FPGA,” in Proc. 1st Int. Conf. Inf. Sci. Eng., Dec. 2009, pp. 200–203. [14] Y. Li and X. Zhang, “Adaptive digital predistortion based on MC-FQRDRLS algorithm using indirect learning architecture,” in Proc. 2nd Int. Conf. Adv. Comput. Control, vol. 4. Mar. 2010, pp. 240–242. [15] G. Montoro, P. L. Gilabert, E. Bertran, A. Cesari, and J. A. Garcia, “An LMS-based adaptive predistorter for cancelling nonlinear memory effects in RF power amplifiers,” in Proc. Asia–Pacific Microw. Conf., Dec. 2007, pp. 1–4. [16] P. L. Gilabert, E. Bertran, G. Montoro, and J. Berenguer, “FPGA implementation of an LMS-based real-time adaptive predistorter for power amplifiers,” in Proc. Joint IEEE North-East Workshop Circuits Syst. TAISA Conf., Jun. 2009, pp. 1–4. [17] F. M. Ghannouchi, O. Hammi, and M. Helaoui, Behavioral Modeling and Predistortion of Wideband Wireless Transmitters, 1st ed. West Sussex, U.K.: Wiley, 2015. [18] N. Kelly and A. Zhu, “Low-complexity stochastic optimization-based model extraction for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 5, pp. 1373–1382, May 2016. [19] J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control, vol. 37, no. 3, pp. 332–341, Mar. 1992. [20] J. C. Spall, “An overview of the simultaneous perturbation method for efficient optimization,” John Hopkins APL Tech. Dig., vol. 19, no. 4, pp. 482–492, 1998. [21] D. Zhou and V. E. DeBrunner, “Novel adaptive nonlinear predistorters based on the direct learning algorithm,” IEEE Trans. Signal Process., vol. 55, no. 1, pp. 120–133, Jan. 2007. [22] L. Guan and A. Zhu, “Dual-loop model extraction for digital predistortion of wideband RF power amplifiers,” IEEE Microw. Compon. Lett., vol. 21, no. 9, pp. 501–503, Sep. 2011. [23] R. N. Braithwaite, “Closed-loop digital predistortion (DPD) using an observation path with limited bandwidth,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 2, pp. 726–736, Feb. 2015. [24] 7 Series FPGAs Overview, Xilinx Inc., San Jose, CA, USA, 2015.

KELLY AND ZHU: DIRECT ERROR-SEARCHING SPSA-BASED MODEL EXTRACTION FOR DPD OF RF PAs

Noel Kelly (S’15) received the B.E. and Ph.D. degrees in electronic engineering from the School of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland, in 2012 and 2017, respectively. His current research interests include lowcomplexity digital predistortion architectures, efficient field-programmable gate array implementation solutions, and digital predistortion applications for satellite communications.

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Anding Zhu (S’00–M’04–SM’12) received the B.E. degree in telecommunication engineering from North China Electric Power University, Baoding, China, in 1997, the M.E. degree in computer applications from the Beijing University of Posts and Telecommunications, Beijing, China, in 2000, and the Ph.D. degree in electronic engineering from the University College Dublin (UCD), Dublin, Ireland, in 2004. He is currently a Professor with the School of Electrical and Electronic Engineering, UCD. His current research interests include high-frequency nonlinear system modeling and device characterization techniques with a particular emphasis on behavioral modeling and linearization of RF power amplifiers for wireless communications, high-efficiency power amplifier design, wireless transmitter architectures, digital signal processing, and nonlinear system identification algorithms.

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A System-on-Chip Crystal-Less Wireless Sub-GHz Transmitter Philipp Greiner, Graduate Student Member, IEEE, Jasmin Grosinger, Member, IEEE, Johannes Schweighofer, Christoph Steffan, Graduate Student Member, IEEE, Sandra Wilfling, Gerald Holweg, and Wolfgang Bösch, Fellow, IEEE Abstract— This paper presents a fully integrated system-onchip wireless transmitter that has no need for an external quartz crystal frequency reference. Instead, a nontrimmable LC oscillator (LCO) is implemented, operating at a fixed frequency of roughly 3.2 GHz. This LCO serves as the accurate frequency reference of the transmitter and is used to derive a frequency in the sub-GHz range for wireless applications via a fractional phase-locked loop. The digital functionality required for the frequency generation and thus for the wireless operation is performed by an 8-b microcontroller. The calibration data as well as the application specific firmware are stored in an integrated EEPROM. An initial frequency accuracy of ±52 ppm over a temperature of −20 °C to 85 °C is achieved using only two temperature insertions for the calibration. The device covers a continuous frequency range of 10–960 MHz and provides an ASK and GFSK operation. A low harmonic power amplifier was implemented to overcome design issues due to injection modulation effects. The chip is implemented in a 130 nm standard CMOS process and constitutes a new approach for a fully integrated wireless sub-GHz transmitter that is competitive to state-of-the-art quartz crystal-based transmitters as well as to already existing crystal-less architectures. Index Terms— Calibration techniques, phase-locked loops (PLLs), RF power amplifiers, RF system-on-chip (SoC) integration, shielding, voltage-controlled oscillator (VCO).

I. I NTRODUCTION

T

YPICAL state-of-the-art wireless sub-GHz architectures use a quartz crystal as an accurate frequency reference [1], [2]. A fractional-N phase-locked loop (PLL) allows the generation of the desired frequency via a variable dividing ratio. The bandwidth of the PLL is often in a range of 50–400 kHz, which allows performance of the frequency modulation via the PLL dividing ratio in a closed-loop configuration [1], [2]. An LC oscillator (LCO) is typically the voltage-controlled oscillator (VCO) of choice as it offers a low phase noise and a low power consumption. While these quartz crystal-based wireless architectures offer the highest Manuscript received December 19, 2016; revised May 23, 2017 and July 4, 2017; accepted August 2, 2017. Date of publication September 14, 2017; date of current version March 5, 2018. This work was supported by the project “Kalium Home Monitoring” funded by the Austrian Research Promotion Agency. (Corresponding author: Philipp Greiner.) P. Greiner, J. Grosinger, and W. Bösch are with the Institute of Microwave and Photonic Engineering, Graz University of Technology, 8010 Graz, Austria (e-mail: [email protected]; [email protected]; wbosch@tugraz). J. Schweighofer, C. Steffan, S. Wilfling, and G. Holweg are with Infineon Technologies Austria AG, 8020 Graz, Austria (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2017.2748130

performance in terms of frequency accuracy and phase noise, they also have some disadvantages. The quartz crystal is an external device that must be combined with two capacitors increasing the number of external components, the pin-count of the wireless chip, and the overall costs. Additionally, the quartz crystal constitutes a fragile mechanical resonator leading to a sensitivity against mechanical shock and vibration. Due to these disadvantages, research efforts are ongoing to avoid quartz crystal-based wireless transmitters. Different research efforts toward a crystal-less wireless transmitter or rather transceiver have led to a variety of solutions. Next to crystal-less architectures that use external frequency references by recovering a clock from a received signal as presented in [3], [4], also architectures with a highly stable frequency reference implemented in the system are possible and are briefly discussed in the following. Microelectromechanical systems (MEMS) oscillator-based solutions lead to a very similar system architecture as in quartz crystalbased transceivers [5], as the frequency of a MEMS oscillator can lie in the same range (tens of megahertz) [5], [6]. However, MEMS oscillators exhibit a stronger temperature dependency than quartz crystal-based oscillators and thus require a temperature compensation [6]. Another possible solution is a crystal-less wireless device based on a bulk acoustic wave (BAW) resonator [7], [8]. These resonators operate at a high frequency in the gigahertz range and as a consequence lead to different system considerations [7]. The quality factor of a BAW resonator is in the range of ∼1000, which allows trimming of its frequency with load capacitors in a small range. Therefore, the necessary temperature compensation and frequency modulation can be performed without a fractional PLL. Nevertheless, recent work has demonstrated a fractional operation of a BAW-based transmitter to enable a variable output frequency and frequency modulation capability [8]. While all of the aforementioned solutions contain some kinds of mechanical resonators to replace the quartz crystal, it is also possible to use a fully integrated CMOS frequency reference. Such a frequency reference comes with the advantages of low costs and small size. Over the past few years, a new type of frequency reference has emerged based on integrated, trimmed, and temperature-compensated LCOs. These oscillators have been demonstrated to operate as highly stable single chip frequency references for quartz crystal replacement [7]–[12]. As these devices achieve a relatively high frequency accuracy in the range of 50–250 ppm as well

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as a low phase noise, the basic requirements can be fulfilled for some areas of the wireless sector. While 2.4 GHz applications as well as some narrowband applications require a high frequency accuracy of ≤20 ppm, most sub-GHz applications require a relaxed accuracy of only about 250 ppm and therefore can potentially be realized with an integrated LCO frequency reference [9]. Several difficulties must be resolved, however, in order to exploit an LCO frequency reference in such an application. For using an LCO as a simple frequency reference for quartz crystal replacement, a calibration to a single frequency is sufficient during the manufacturing process. In contrast, for a wireless application, a field programmability of the operating frequency is available via the fractional-N PLL in state-of-the-art quartz crystalbased wireless devices [1], [2] and is therefore considered a basic requirement in this paper. In general, state-of-theart LCO-based CMOS frequency references do not offer this feature [10], [11]. Furthermore, state-of-the-art LCO frequency references operate at gigahertz frequencies and use capacitive trimming in combination with an integer frequency divider to cover a wide output frequency range [11], [12]. Trimming capacitors are expected to degrade the LC tank in terms of its quality factor (Q), leading to a reduced phase noise performance. Additionally, trimming capacitors introduce nonlinearities to the LC tank, which is considered particularly critical as it makes the LCO frequency sensitive to the LCO amplitude, reducing the frequency accuracy. Consequently, a high integer dividing ratio is typically used to keep the required trimming range small, e.g., 3% [12]. The output frequency of state-of-the art LCO frequency references is thus limited to a maximum value of approximately 200 MHz [10]. For a higher output frequency, the trimming range has to be extended accordingly leading to a higher portion of trimming capacitors. The only currently known all CMOS single chip wireless transmitter that solves the above mentioned difficulties is the Silicon Laboratories SI4010 chip [9]. It is based on a trimmed LCO in combination with a frequency divider and enables a simplified wireless architecture with a minimum number of external devices for applications in the sub-GHz range. To offer a field programmability of the operating frequency, the SI4010 uses a sophisticated self-calibration process that must be performed prior to every data transmission, leading to additional energy consumption. In contrast to previous work, this paper presents a new approach for a crystal-less wireless sub-GHz transmitter. It is based on a nontrimmable LCO as the frequency reference [13] from which a variable and temperature-compensated RF carrier signal can be derived for a wireless transmitter operation. Greiner et al. [13] have introduced a new architecture for an LCO frequency reference. This architecture is based on a nontrimmable LCO as the reference in combination with a high-resolution fractional interpolating frequency divider. By performing a calibration, temperature compensation, and frequency setup via the fractional dividing ratio, this architecture inherently provides a field programmability of the frequency without the degradation of Q or the linearity by trimming capacitors.

Fig. 1. Simplified block diagram of the SoC crystal-less wireless transmitter chip: the analog core comprises an accurate all CMOS frequency generator allowing operation of the device as a sub-GHz transmitter using the PA or as a frequency reference in the range of up to 180 MHz using the CLK driver output. The digital core performs the temperature compensation as well as the ASK and FSK modulation using the calibration data and the application specific firmware stored in the EEPROM.

This paper is organized as follows. Section II presents the overall transmitter system and its functional blocks. Section III describes the system implementation with emphasis on the frequency generation as well as the adaptation of the system architecture with respect to state-of-the-art wireless sub-GHz transmitters. Experimental results comprising calibration and performance characteristics as well as a comparison with previous work are presented in Section IV. II. S YSTEM OVERVIEW Fig. 1 depicts the simplified block diagram of the system. The analog core includes the all CMOS frequency reference that is created by the nontrimmable LCO in combination with the fractional interpolating frequency divider. The analog core also includes a temperature sensor for a temperature compensation as well as the PLL to multiply the output frequency of the fractional divider. Furthermore, an analog power management unit (PMU) is implemented consisting of a bandgap reference and several voltage regulators for the respective analog blocks. The digital core is based on an 8-b microcontroller and offers all the functionalities needed for the generation of an accurate and temperature-compensated frequency and the wireless data transmission. The digital core thus consists of digital low-pass filters (LPFs) for the temperature sensor and an ASK/frequency-shift keying (FSK) modulator block. A ROM is included, which contains precast firmware functions for the temperature compensation. An EEPROM is included for calibration data as well as an application specific firmware. After a reset, the application specific firmware is loaded into the SRAM from where it can be executed during the operation. A fractional-N baud rate divider is used to generate an accurate baud rate derived from the all CMOS frequency generator. The microcontroller clock is provided by an integrated

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Fig. 3. Simplified architecture of the LCO: the LCO is arranged for a symmetric operation around half of the supply voltage using a pair of cross coupled nMOS and pMOS transistors, respectively. Additionally, an AAC is applied in order to operate the LCO at an optimum voltage amplitude. Fig. 2. Micro photography of the crystal-less wireless transmitter: the chip is produced in a 130 nm standard CMOS process.

24 MHz RC oscillator. Furthermore, a serial peripheral interface for configuration and several general-purpose input/output pins are implemented. The digital PMU contains the power-on reset circuitry as well as a low power voltage regulator for the digital core. A PA is implemented as an output stage, specifically designed for the requirements of this system operating as a wireless transmitter. Additionally, a CMOS driver output stage is available, which allows to operate the device as a frequency reference in single ended or differential mode. For the layout design, different considerations have been taken into account. Electromagnetic interactions are very critical as the output stages can produce a high amount of harmonics. Also, the digital core can produce distortions. In order to minimize coupling effects, a magnetic shield has been implemented, which consists of a metal ring closely surrounding the inductor. Additionally, the supply domains were separated in order to minimize interferences to the sensitive analog core. Furthermore, heat dissipation and the corresponding temperature gradients can lead to considerable temperature differences across the chip. For an accurate frequency generation, it is mandatory to measure the unaltered temperature of the LCO. Therefore, the temperature sensor has been placed close to the LCO, while the major heat dissipaters (i.e., voltage regulators and PA) have been placed at the chip edge to realize a maximum distance to the LCO. Fig. 2 displays the micro photography of the produced chip and the location of the most important functional blocks. III. S YSTEM I MPLEMENTATION A. Crystal-Less Wireless Architecture For a crystal-less wireless operation, an architecture has been implemented, which allows the generation of an RF carrier signal with a high frequency accuracy. Also, a modulation of this RF carrier is required for a wireless transmitter. An accurate reference frequency is therefore generated using

Fig. 4. Simplified block diagram of the fractional interpolating frequency divider: an accumulator-based fractional N/N + 1 frequency divider is used in order to generate an accurate fractional dividing ratio. The corresponding phase error is corrected by means of an additional phase interpolator.

the nontrimmable LCO and the fractional interpolating frequency divider. Fig. 3 shows the simplified architecture of the LCO. The LCO is designed for a symmetric operation around half of the core supply voltage. An automatic amplitude control (AAC) is employed to adjust the LCO amplitude to a variable value using a programmable reference voltage. The AAC allows us to adjust the LCO amplitude to an optimum value in order to minimize the frequency sensitivity. As a result, frequency errors can be minimized, which are caused by drifts of this amplitude over the device lifetime. Fig. 4 shows the simplified block diagram of the fractional interpolating frequency divider. In order to achieve a precise fractional dividing ratio, an accumulator-based fractional N/N + 1 divider is used. The phase error that is caused by the fractional N/N + 1 architecture is corrected using a phase interpolator. The phase interpolator offers a high time resolution of about 10 ps. A final division by two assures an accurate rising and falling edge of the generated reference frequency, allowing it to be used for the clock (CLK) driver output stage. Fig. 5 shows the functional block diagram of the RF carrier generation and modulation. An accurate reference frequency is generated by means of a defined dividing ratio and is then scaled to the desired RF carrier frequency using the PLL. The modulator can be used for FSK as well as ASK operation. It is designed to offer slope- and Gaussian-shaped transitions for ASK, FSK, and GFSK and can operate with

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Fig. 6. Simplified block diagram of the signal processing chain for the temperature compensation comprising the sigma–delta temperature sensor followed by the second-order digital LPF, the compensation polynomial, and the frequency setup (realized via firmware) to provide the correct dividing ratio. The second LPF stage is implemented to provide an overall third-order low-pass behavior. Fig. 5. Simplified architecture for the frequency generation and modulation: an accurate RF carrier is generated by the all CMOS frequency generator in conjunction with the PLL. Modulation can be performed via the dividing ratio of the fractional divider for FSK or via the PA for ASK.

NRZ and Manchester coding. The modulator thus requires a clock with an oversampling ratio of 16 times the baud rate that is generated by the baud rate divider. The baud rate divider uses the accurate reference frequency provided by the fractional divider and generates a baud rate with a resolution of better than 0.05 % via a fractional-N architecture. For FSK operation, the modulation of the frequency is achieved by altering the dividing ratio of the fractional divider as it allows a high reconfiguration rate. The modulator block thus produces a digital modulation signal that is then added to the dividing ratio. In this way, an accurate and modulated reference frequency for FSK operation is realized. For ASK operation, only a fixed reference frequency is used and the modulation is performed via the PA. B. Temperature Compensation For integrated LCOs, a considerable dependency of the frequency on the temperature has been shown in previous publications [12], [14]. For example, uncompensated linear temperature coefficients of 20 and −268 ppm/K have been reported in [12] and [14], respectively. A temperature compensation is thus necessary in order to obtain an accurate reference frequency. For this reason, a temperature sensor is implemented on this chip. The temperature sensor uses an architecture similar to [15]. Two substrate p-n-p transistors biased with different current densities are used to generate a voltage that is proportional to the absolute temperature (PTAT) as well as a temperature-independent bandgap voltage VBG . A digital temperature value is generated by converting the PTAT voltage via a second-order sigma–delta analog-to-digital converter (ADC) while VBG is used as the reference. Fig. 6 depicts a simplified block diagram of the signal processing chain for the temperature compensation. For further processing of the temperature, the 1-b ADC output signal is low-pass filtered using a second-order digital infinite impulse response (IIR) filter. While this digital value can be used to compensate the temperature characteristic of the LCO, any overlaying noise leads to a modulation of the frequency and as a consequence to phase noise. This effect is particularly critical as the sigma delta ADC produces a high amount of quantization noise. The cut-off frequency of the second-order IIR filter must thus be chosen accordingly low. Measurement

results show that up to a cut-off frequency of 400 Hz, the phase noise due to quantization noise is lower than the phase noise produced by the LCO, and consequently no degradation occurs. While a 400 Hz cut-off frequency corresponds in a sufficiently fast temperature response, the initial settling time after turning the device ON will correspond in an unnecessarily long start-up latency of up to 20 ms. To overcome this issue, the second-order IIR filter is implemented with a variable cut-off frequency. In the first few milliseconds, the filter is adjusted to a high cut-off frequency of 3.2 kHz to achieve a fast settling time and is then switched to 400 Hz before the LCO is turned ON. This implementation leads to a fast startup within 5 ms while not degrading the phase noise performance. Simulation results as well as a previous publication [11] imply that the LCO has a distinct second-order temperature behavior. Higher order effects also occur but are less pronounced. A forth-order polynomial has thus been implemented as part of the compensation firmware, which is applied to the output signal of the first LPF stage. The compensated value constitutes a raw dividing ratio for a temperatureindependent frequency. This value is subsequently multiplied with a frequency setup value to obtain the correct dividing ratio for the desired operating frequency. The frequency setup value is a digital value referable to a periodic time, as a multiplication is easier to implement than a division in the implemented microcontroller. The compensation firmware is cyclically computed during the frequency generation with a sampling rate of several kilohertz. Finally, the dividing ratio is low-pass filtered with a further digital IIR filter to achieve a third-order overall low-pass characteristic. C. Ring-Oscillator PLL The PLL is used to enable generating higher frequencies to cover the whole sub-GHz frequency range, as the output frequency of the fractional divider is limited to 180 MHz. Fig. 7 depicts the block circuit diagram of the PLL. The PLL comprises a frequency doubler at its input allowing operation at an effective reference frequency of up to 360 MHz. A fast phase frequency detector is realized with a minimum turn on time of 300 ps to avoid dead zone errors. For a variable loop characteristic, a charge pump is implemented offering an adjustable current as well as a loop filter with a widely configurable loop bandwidth. The VCO in this PLL is realized by means of a ring oscillator. A PMOS transistor is used to convert the output voltage of the loop filter into the supply current of the ring oscillator. The VCO is followed by a power

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Fig. 7. Block circuit diagram of the ring-oscillator PLL: the ring-oscillator frequency can be variably divided in power of two steps via the PLL MUX and the OP MUX to provide the PLL feedback and the RF signal within a wide frequency range.

of two scalable frequency divider that is used to generate the scaled PLL feedback frequency as well as the desired RF output frequency via the PLL multiplexer (PLL MUX) and the output multiplexer (OP MUX), respectively. The ring oscillator is designed for an operating frequency range of 1.3–2.6 GHz and for a current consumption of about 1 mA at an operating frequency of 1.736 GHz for a divided output frequency of 868 MHz. Using a ring oscillator as the VCO comes with the advantage of a small chip area occupation and unlike LCO-based VCOs with a low sensitivity to magnetic interactions. The phase noise of a ring oscillator is substantially worse in comparison with an LCO at low offset frequencies, however, while the phase noise at high offset frequencies can be sufficiently low for sub-GHz applications. Accordingly, the PLL must be operated with a high loop bandwidth. D. Power Amplifier During the course of the transmitter design, it was found that under certain conditions, an unwanted modulation of the LCO frequency occurs due to harmonic distortions of the output stage and that this modulation is a critical problem for the transmitter. This effect, referred to as injection pulling/modulation, is caused by harmonics in the current of the output stage or the PA that lie exactly at/or very close to the LCO frequency. Based on magnetic coupling to the LCO inductor, these harmonics can lead to frequency errors for odd integer ratios between the LCO and the output frequency or to a modulation for ratios very close to odd integer values. As a result, injection pulling can lead to a shift of the carrier frequency or to spurious tones in the spectrum, which cause issues with respect to an error-free reception and with regulation bodies. There is a high probability for this effect to occur due to the fact that many different frequencies are available in the sub-GHz range and due to the fact that the nontrimmable LCO frequency varies due to process variations. Injection pulling/modulation also occurs in crystal-based transmitters with LCO-based VCOs [16], yet is less critical in a PLL configuration. If the realized chip is operated as a frequency reference, this effect can be neglected up to 180 MHz using a limited slew rate of the output stage for typical impedance values of the load. However, a high slew rate is inevitable for higher output frequencies up to 1 GHz.

Fig. 8. Block circuit diagram of the PA: a differential current architecture with Gaussian-shaped transitions is used to reduce harmonics and interferences.

In order to reduce the injection pulling/modulation effect, a differential current-based PA architecture has been chosen for implementation within this paper. As a result, the current is mostly influenced by the PA, while the load impedance has only a minor influence on the current transition shape. A differential current-based architecture of the PA is also implemented in [9]. Fig. 8 shows the simplified block circuit diagram of the PA implemented in this paper. The output stage of the PA is implemented as a differential open drain stage powered by a programmable current source against VSS. The load must consequently constitute a differential impedance with respect to VDD. To reduce the voltage drop across the transistors, an additional cascode is implemented. With this approach, a constant dc current trough the PA is achieved, which thus produces little interference. Considerable interferences can in any case also be induced by harmonics from the differential current of the PA. This harmonics have been reduced by means of Gaussian shaping of the current transitions. A delay line is thus used, which generates five phasings for the RF carrier signal of the positive and negative output current path, respectively. The five phasings are recombined via a Gaussian weighted resistor array, leading to Gaussian-shaped voltage transitions on the differential current output stage. The shaped voltage signal is converted into the output current by the output stage. The gate capacitance in conjunction with the resistor array thereby acts as an additional LPF. In addition to counteracting injection pulling/modulation effects, a reduced amount of harmonics also simplifies the design of the antenna filter/matching network. A drawback of this implementation is that the Gaussian-shaped current transitions lead to a slightly degraded PA efficiency in comparison with rectangular-shaped output currents. For a tradeoff between PA efficiency and harmonic distortion at the respective operating frequency, the delay line is adjustable corresponding in a rise/fall time in the range between 250 ps and 1.5 ns. IV. E XPERIMENTAL R ESULTS A. Calibration and Frequency Accuracy To provide an accurate frequency reference, every single device must be calibrated with respect to the temperature.

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Fig. 9. Initial frequency error of 28 devices calibrated with two temperature insertions at 0 °C and 70 °C. The black lines indicate the average error as well as the ±3σ limits. An initial accuracy of ±52 ppm is achieved over a temperature range of −20 °C to 85 °C.

The temperature curve of the frequency has thus been measured for eight devices to determine an averaged third-order compensation polynomial optimized for the temperature range between −20 °C and 85 °C. A linear temperature dependency of −68 ppm/K was found at 27 °C with overlaying higher order effects. Additionally, a significant correlation between the linear and higher order coefficients was found and allows linear scaling of these coefficients. By exploiting this correlation, only two temperature insertions (at 0 °C and 70 °C) are required to determine the constant term as well as the linear and higher order coefficients of the polynomial for every single device. The initial frequency accuracy that has been achieved with this calibration approach is depicted in Fig. 9 showing the initial frequency accuracy versus temperature for 28 devices. Although a high initial accuracy of ±52 ppm (3σ ) can be achieved, additional frequency errors can occur over the device lifetime due to ageing, moisture, and mechanical stress. Also, a frequency sensitivity with respect to the supply voltage of about 2.6 ppm/V has been observed over the supply voltage range from 1.8 to 3.6 V. The measured devices have been taken from one wafer. Therefore, this calibration method could be sensitive to process variations. B. Phase Noise Performance Fig. 10 shows the measured single-sideband (SSB) phase noise power spectral density (PSD) of the presented device at a carrier frequency of 868 MHz using a reference frequency of 217 MHz. The loop bandwidth of the PLL has been adjusted to 3.2 MHz for an optimal performance. At low offset frequencies, the phase noise is determined by the LCO phase noise, while for higher offset frequencies, the fractional divider produces additional noise in the form of a higher noise floor. Also, additional measurements showed that spurious tones can occur due to the fractional-N architecture at a level of −52 dBc (measured at 434 MHz). C. Power Amplifier/Injection Pulling For the experimental verification of the PA, a matching network has been realized, which is suitable for the 50  test equipment and provides a differential load of approximately 430 . A maximum transmit power of 9.5 dBm has been achieved with this 430  load for an operating frequency

Fig. 10. Phase noise measured at a carrier frequency of 868 MHz: for different offset frequencies, the phase noise is determined by the LCO phase noise, the fractional divider, and the ring oscillator, respectively. For comparison reasons, the datasheet typical values for the phase noise of the Silicon Labs SI4010 [9] device and the Infineon TDA5150 [2] device are shown.

Fig. 11. Transient plot of the frequency deviation from the 1.06 GHz carrier obtained by means of a frequency demodulator: for a dividing ratio of 3.000000059 a modulated and for a dividing ratio of exactly three an unmodulated carrier signal can be observed. Due to injection pulling, a maximum frequency deviation of 6.2 kHz is observed corresponding to a frequency error of 5.8 ppm.

of 434 MHz. This maximum power can be maintained for a supply voltage down to 2.1 V and is then continuously decreasing to a value of 6.5 dBm at the minimum supply voltage of 1.8 V. In order to test the sensitivity to injection pulling effects, the transmitter is operated with the worst case odd integer dividing ratio of three corresponding to a frequency of approximately 1.06 GHz. For the sensitivity test, the closest digital dividing ratio next to three has been chosen resulting in a frequency deviation of 20 Hz. As a result of injection pulling, the output frequency is modulated with 60 Hz, i.e., three times the frequency deviation. The corresponding modulated frequency for the dividing ratio very close to three as well as the unmodulated carrier for the dividing ratio of exactly three is depicted in Fig. 11. A maximum frequency error referable to injection pulling of 5.8 ppm has been observed under worst case conditions using the highest possible frequency, the maximum current for the PA, and the minimum PA current transition time of 250 ps. In practice, an odd integer dividing ratio is never obtained for the frequency range between 868 and 960 MHz considering the frequency variations of the LCO. Also for lower frequencies of up to 434 MHz, a higher PA current transition time can be used, so that injection pulling effects can be neglected.

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TABLE I C OMPARISON OF T RANSMITTER K EY P ERFORMANCE C HARACTERISTICS

Fig. 12. Transient plot of a data transmission event: the different states of operation as well as the current consumption are plotted for an ASK operation. Also, the RF-output signal is shown for an optimal matching at a 50  load.

The comparison of the transmitter using the presented PA with a similar previous test chip with a conventional class-C PA, as used in [2], shows a reduction of injection pulling effects by 20 and 18 dB for a dividing ratio of seven (close to 434 MHz) and three (close to 868–960 MHz), respectively. It must be noted that these results can vary greatly, as for a class-C PA, the harmonics of the output current are strongly influenced by the external matching and filtering components. D. Transmitter Operation To demonstrate the functionality of the transmitter, a simple arrangement of a 434 MHz ASK transmitter has been realized on a test board. Fig. 12 shows the transient graph for one transmitted data package. A baud rate of 5 kb/s and Manchester coding are used. The current consumption is also shown in Fig. 12 in relation to the respective state of operation. At the beginning of the data transmission, the chip is started up from the deep sleep mode by an external interrupt. The firmware is then loaded from the EEPROM and subsequently executed. For this arrangement, the temperature sensor filters are settled for 6.5 ms. During the temperature settling time, the current consumption is low with about 1.2 mA. Subsequently, the LCO and the PLL are turned ON. The PLL lock is typically achieved within less than 10 μs and is then followed by the transmission of the data package. The overall current consumption for the RF carrier generation including the continuous temperature compensation amounts to about 8 mA. The transmit current consumption depends on the adjusted output power and can range up to 18.5 mA for the maximum power of 9.5 dBm (as shown in Fig. 12). After the data transmission, the transmitter goes back to the deep sleep mode in which it consumes 500 nA. Using this test arrangement, compatibility with stateof-the-art sub-GHz receivers has been demonstrated in the lab with different baud rates for ASK, FSK, and GFSK modulation. E. Performance Comparison The key performance characteristics of this paper are compared in Table I with the Silicon Labs crystalless system-on-chip (SoC) RF transmitter Si4010 [9] and

the quartz crystal-based Infineon state-of-the-art transmitter TDA5150 [2]. While the Si4010 as well as this paper are crystal-less SoC transmitter implementations, the TDA5150 is a single transmitter chip requiring an external microcontroller. For the crystal-less operation, new system architectures are exploited for both crystal-less SoC devices based on integrated LCOs. The field programmability of the operating frequency is realized by capacitive frequency trimming in the SI4010 and by a fractional PLL in this paper. In comparison with quartz crystals, one goal of the design of a crystal-less SoC device is to achieve a minimum number of external components and consequently a simple and costefficient overall system. The numbers of components that are presented in the datasheet application examples are compared in Table I neglecting the battery, switches, and the microcontroller for the TDA5150. Also, the antenna is not included as it can be implemented on the PCB. The TDA5150 requires seven external components. The SI4010 requires only two external components using a PCB antenna. A simple PCB including an antenna has been developed in this work also leading to only two external components (i.e., two capacitors). A high initial frequency accuracy of ±52 ppm has been achieved in this paper. However, due to ageing, an additional frequency error is expected over lifetime. Standard reliability tests have been performed, including HAST, HTOL, and one solder reflow test, resulting in an overall frequency accuracy of 205 ppm. For the TDA5150, the frequency accuracy is given by the crystal and can therefore be much better (i.e., 10–100 ppm). For comparison reasons, the datasheet values of the phase noise are plotted for the SI4010 as well as for the TDA5150 in Fig. 10. At an offset frequency of 10 kHz, a low phase noise of −80 dBc/Hz is achieved in this paper, which is comparable with the quartz crystal-based transmitter TDA5150. The significantly lower phase noise in comparison to the LCO-based transmitter SI4010 is assumed to be attributable to the nonexistent lossy trimming components in this paper. In the range between 10 kHz and 1 MHz, a phase noise performance is achieved, which is similar to the SI4010 and significantly better than the TDA5150. For offset frequencies higher than 1 MHz, this paper achieves the worst phase noise performance, which is caused by the high phase noise of the ring

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oscillator-based VCO as well as the required high loop bandwidth. In this paper, a power consumption similar to the SI4010 has been achieved despite the more complex system with a fractional divider and a PLL. However, in comparison with the quartz crystal-based transmitter, the current consumption is higher. The high current consumption results due to the RF carrier generation based on the LCO architecture and due to the transmit operation since a less efficient differential current PA is used. The performance comparison shows that the crystal-less transmitter presented in this paper constitutes a new approach for a fully integrated wireless sub-GHz transmitter that is competitive to state-of-the-art quartz crystal-based transmitters as well as to already existing crystal-less architectures.

[10] Si500S Single-Ended Output Silicon Oscillator Rev. 1.1, Silicon Lab. Inc., Austin, TX, USA, Oct. 2011. [11] M. S. McCorquodale and V. Gupta, “A history of the development of CMOS oscillators: The dark horse in frequency control,” in Proc. Joint Conf. IEEE Int. Freq. Control Eur. Freq. Time Forum (FCS), May 2011, pp. 1–6. [12] A. Marques, “Reference-less clock circuit,” U.S. Patent 7 332 975 B2, Feb. 19, 2008. [13] P. Greiner, J. Grosinger, C. Steffan, G. Holweg, and W. Bösch, “Non-trimmable LC oscillator for all CMOS frequency control,” in Proc. 41st Eur. Solid-State Circuits Conf. (ESSCIRC), Sep. 2015, pp. 140–143. [14] M. S. McCorquodale et al., “A 25-MHz self-referenced solid-state frequency source suitable for XO-replacement,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 5, pp. 943–956, May 2009. [15] M. A. P. Pertijs, K. A. A. Makinwa, and J. H. Huijsing, “A CMOS smart temperature sensor with a 3σ inaccuracy of ±0.1 °C from −55 °C to 125 °C,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2805–2815, Dec. 2005. [16] B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1415–1424, Sep. 2004.

V. C ONCLUSION A new approach for a crystal-less wireless sub-GHz transmitter has been presented. The approach is based on an integrated nontrimmable LCO and a fractional ring oscillator PLL, and therefore achieves the highest possible level of integration with a minimum number of external devices. Sufficiently low phase noise has been demonstrated for sub-GHz wireless applications. However, the lower frequency accuracy of the integrated LCO frequency reference is considered the major drawback with respect to a quartz crystalbased solution. In this paper, a power consumption similar to the SI4010 and slightly higher than in state-of-the-art quartz crystal-based devices has been achieved. For future work, a similar approach with a sigma–delta fractional divider could help to obtain a power consumption closer to the quartz crystal-based devices. In addition, a further development of a fully integrated SoC crystal-less transceiver is feasible as the transmit functionality demonstrated in this paper is considered to be more critical and thus no obstacles are expected for the implementation of a receiver. R EFERENCES [1] P. Crowley et al., “A sub 1 GHz versatile CMOS wireless transceiver,” in Proc. IEE Seminar Telemetry Telematics, Apr. 2005, pp. 7-1–7-7. [2] TDA5150 Multichannel/Multiband Transmitter Rev. 1.1, Infineon Technol. AG, Neubiberg, Germany, Jun. 2012. [3] G. Papotto, F. Carrara, A. Finocchiaro, and G. Palmisano, “A 90-nm CMOS 5-Mbps crystal-less RF-powered transceiver for wireless sensor network nodes,” IEEE J. Solid-State Circuits, vol. 49, no. 2, pp. 335–346, Feb. 2014. [4] H. Bhamra et al., “A 24 μW, batteryless, crystal-free, multinode synchronized SoC ‘Bionode’ for wireless prosthesis control,” IEEE J. Solid-State Circuits, vol. 50, no. 11, pp. 2714–2727, Nov. 2015. [5] G. Chance et al., “Integrated MEMS oscillator for cellular transceivers,” in Proc. IEEE Int. Freq. Control Symp. (FCS), May 2014, pp. 1–3. [6] K. L. Phan et al., “High precision frequency synthesizer based on MEMS piezoresistive resonator,” in Proc. 17th Int. Conf. Solid-State Sens., Actuators Microsyst. (TRANSDUCERS EUROSENSORS), Jun. 2013, pp. 802–805. [7] M. Flatscher et al., “A bulk acoustic wave (BAW) based transceiver for an in-tire-pressure monitoring sensor node,” IEEE J. Solid-State Circuits, vol. 45, no. 1, pp. 167–177, Jan. 2010. [8] R. Thirunarayanan, D. Ruffieux, N. Scolari, and C. Enz, “A  based direct all-digital frequency synthesizer with 20 Mbps frequency modulation capability and 3 μs startup latency,” in Proc. 41st Eur. SolidState Circuits Conf. (ESSCIRC), Sep. 2015, pp. 388–391. [9] Si4010-C2 Crystal-Less SoC RF Transmitter Rev. 1.0, Silicon Lab. Inc., Austin, TX, USA, Feb. 2011.

Philipp Greiner (GS’15) was born in Graz, Austria, in 1986. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Graz University of Technology, Graz, in 2011 and 2013, respectively. His graduate research focused on NFC-RFID with an emphasis on analog integrated circuits. He is currently pursuing the Ph.D. degree in wireless miniaturization (with a special focus on highly stable all CMOS frequency generation for crystal quartz replacement).

Jasmin Grosinger (S’09–M’12) received the Dipl.Ing. (M.Sc.) degree (Hons.) in telecommunications and the Dr.Techn. (Ph.D.) degree (Hons.) from the Vienna University of Technology, Vienna, Austria, in 2008 and 2012, respectively, where she examined backscatter radio frequency systems and devices for novel wireless sensing applications. From 2008 to 2013, she was a Project Assistant with the Institute of Telecommunications, Vienna University of Technology, where she was involved in various projects dealing with RFID technologies. In 2011, she was a Lab Associate with Disney Research, Pittsburgh, PA, USA, involved in backscatter RFID sensors. Since 2013, she has been a Post-Doctoral Researcher with the Institute of Microwave and Photonic Engineering, Graz University of Technology, Graz, Austria, and heads the Research Group RFID Technologies. She has authored or co-authored more than 30 peer-reviewed publications and holds 1 U.S. patent. Dr. Grosinger is actively involved in Technical Program Committees of various RFID-related conferences and is an Associate Editor of the IET Microwaves, Antennas, and Propagation journal. In addition, she is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), a Management Committee member of the EU COST action IC1301 on wireless power transmission for sustainable electronics, and a member of the Union Radio-Scientifique Internationale Austria, Commission D. She also serves as a member of the IEEE MTT-S RFID Technologies Committee (MTT-24).

Johannes Schweighofer was born in Weiz, Austria, in 1983. He received the bachelor’s and master’s degrees (DI) in information and computer engineering (Telematik) from the Graz University of Technology, Graz, Austria, in 2007 and 2009, respectively. He was a Project Associate with the Institute of Electronics, Graz University of Technology, involved in the chip design and the embedded firmware for an intelligent subgigahertz wireless transceiver as part of an EU-funded project (CHOSeN, FP7). In 2011, he joined Infineon Technologies Austria AG, Graz, where he is involved in various industrial research projects focusing on ultralow power contactless (RFID), wireless (subgigahertz communication), and sensor technologies.

GREINER et al.: SOC CRYSTAL-LESS WIRELESS SUB-GHz TRANSMITTER

Christoph Steffan (GS’15) was born in Graz, Austria, in 1986. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Graz University of Technology, Graz, in 2011 and 2013, respectively. His graduate research focused on dc to dc converters (with an emphasis on analog integrated circuits). He is currently pursuing the Ph.D. degree in integrated energy harvesting interfaces (with a special focus on ultralow power designs for autonomous wireless sensor nodes).

Sandra Wilfling was born in Feldbach, Austria, in 1994. She received the B.Sc. degree in information and computer engineering from the Graz University of Technology, Graz, Austria, in 2016, and is currently pursuing the master’s degree in information and computer engineering at the Graz University of Technology.

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Gerald Holweg was born in Graz, Austria, in 1960. He received the master’s degree (DI) in electronic engineering from the Graz University of Technology, Graz, in 1983. He was an ASIC Design Engineer with AMIAustria, in 1984, specializing in the areas of process parameter extraction, critical analog ASIC design, mixed analog/digital design, chip layout optimization, and the design of telecom circuits. In 1987, he joined the startup company MIKRON-Austria as a Project Manager for RFID ASICS and Subsystems and assumed section management of an RFID design group in 1991. Since 1993, he was responsible for the definition and development of the worldwide first chip and coil contactless smart card (MIFAREÂ). In 1995, he was the Development Manager for the product line contactless smart cards with MIKRON, which joined PHILIPS in 1995. In 1998, he was the Director of Development for Chip Card and Security IC’s with the startup Design Centre SIEMENS Entwicklungszentrum für Mikroelektronik, Graz, which became the INFINEON Technologies Development Centre in 1999. Since 2003, he has been responsible for predevelopment programs and industrial research projects. Wolfgang Bösch (F’13) received the engineering degree from the University of Technology, Vienna, Austria, the Engineering degree from the Universities of Technology, Graz, Austria, and the M.B.A. degree (with distinction) from the Bradford University School of Management, Bradford, MA, USA, in 2004. He was the CTO with the Advanced Digital Institute, U.K., a not-for-profit organization to promote research activities. Earlier, he served as the Director of Business and Technology Integration with RFMD, U.K. He has been with Filtronic plc, as CTO of Filtronic Integrated Products and the Director of the Global Technology Group for nearly 10 years. Before joining Filtronic, he held positions with the European Space Agency, where he was involved in amplifier linearization techniques; MPRTeltech, Canada, where he was involved in MMIC technology projects; and the Corporate Research and Development Group, M/A-COM, Boston, MA, USA, where he was involved in advanced topologies for high-efficiency power amplifiers. In 2010, he joined the Graz University of Technology, to establish a new Institute for Microwave and Photonic Engineering. For four years, he was with DaimlerChrysler Aerospace (now Airbus), Germany, where he was involved in T/R modules for airborne radar. He was a Nonexecutive Director of Diamond Microwave Devices with the Advanced Digital Institute. He is currently a Nonexecutive Director with the VIPER-RF Company, U.K. He has authored more than 80 papers and holds 4 patents. Mr. Bösch is a Fellow of the IET.

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1

Digital Predistortion for Multi-Antenna Transmitters Affected by Antenna Crosstalk Katharina Hausmair, Per N. Landin, Ulf Gustavsson, Christian Fager, and Thomas Eriksson

Abstract— In this paper, a digital predistortion (DPD) technique for wideband multi-antenna transmitters is proposed. The proposed DPD compensates for the combined effects of power amplifier (PA) nonlinearity, antenna crosstalk, and impedance mismatch. The proposed technique consists of a linear crosstalk and mismatch model block shared by all transmit paths and a dual-input DPD block in every transmit path. By avoiding the use of multi-input DPD blocks in every transmit path, the complexity of the proposed technique is kept low and scales more favorably with the number of antennas than competing techniques. It is shown that all blocks can be identified from measurements of the PA output signals using least-squares estimation. Measurement results of a four-path transmitter are presented and used to evaluate the proposed DPD technique against existing techniques. The results show that the performance of the proposed DPD technique is similar to those of existing techniques, while the complexity is lower. Index Terms— Antenna crosstalk, digital predistortion (DPD), multiple-input multiple-output (MIMO) transmitter, power amplifier (PA) linearization.

I. I NTRODUCTION

M

ULTI-ANTENNA systems are an important part of modern and future wireless telecommunication standards such as LTE, Wi-Fi, and 5G [1]. In such systems, each transmit path has its own power amplifier (PA) and antenna element, as shown in Fig. 1. Large-scale multi-antenna systems like massive multiple-input multiple-output (MIMO) comprise up to several hundreds of transmit paths [2]. Therefore, integrated system designs are used where expensive and bulky components like isolators between PAs and antennas are avoided to reduce system complexity and cost. However, Manuscript received March 20, 2017; revised May 23, 2017 and July 19, 2017; accepted August 4, 2017. This research has been carried out in the GigaHertz Centre in a joint project financed by the Swedish Governmental Agency for Innovation Systems (VIN-NOVA), Chalmers University of Technology, Ericsson, Infineon Technologies Austria, Ampleon, National Instruments, Gotmic, and Saab. This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. (Corresponding author: Katharina Hausmair.) K. Hausmair and T. Eriksson are with the Department of Electrical Engineering, Communication Systems Group, Chalmers University of Technology, 41296 Göteborg, Sweden (e-mail: [email protected]; [email protected]). P. N. Landin is with Ericsson Supply Kumla, 692 33 Kumla, Sweden (e-mail: [email protected]). U. Gustavsson is with Ericsson AB Research, SE-402 78 Göteborg, Sweden (e-mail: [email protected]). C. Fager is with the Department of Microtechnology and Nanoscience, Microwave Electronics Laboratory, Chalmers University of Technology, 41296 Göteborg, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2748948

Fig. 1. Multi-antenna transmitter system model with K transmit paths. Each path consists of one PA connected to an antenna element. The antenna elements form the antenna array.

such systems are vulnerable to antenna crosstalk due to mutual coupling and antenna mismatches [3]. As a consequence, integrated multi-antenna transmitters typically suffer from nonlinear distortion due to the mixing of the antenna crosstalk and mismatch with the PA output, in addition to the nonlinear distortion caused by the behavior of the PAs [4], [5]. To avoid violating spectrum regulations and communication standard requirements, compensation techniques are needed to mitigate this distortion at the transmitter. Digital predistortion (DPD) has been widely used to linearize PAs. Many algorithms have been proposed for singleinput DPDs (see [6], [7]), which are designed for systems with only one transmit path. Single-input DPD can compensate for PA nonlinearity, but not for the effects caused by antenna crosstalk, and is therefore not suitable for multi-antenna systems [4], [5]. Several DPD techniques have been proposed to compensate for different types of nonlinear distortion in multi-antenna transmitters. Suryasarman et al. [8] and Suryasarman and Springer [9] and [10] propose a DPD technique that has a structure specifically designed to compensate for nonlinear distortion in systems where crosstalk is introduced before the PA. A similar technique is used in [11]. However, nonlinear effects of antenna crosstalk are not considered and cannot be compensated for using these DPD techniques. The crossover DPDs proposed in [12] and [13] are designed to compensate for crosstalk before and after the PA. However, in those publications, it is assumed that only crosstalk before the PAs is causing nonlinear effects, whereas the effects of crosstalk

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occurring after the PAs are assumed to be linear. Since the techniques do not consider necessary crossterms between signals of different transmit paths, they are not suitable for linearization of multi-antenna transmitters with antenna crosstalk. Abdelhafiz et al. [14] include certain crossterms between different transmit paths in their augmented crossover DPD. However, also their system model does not consider mixing between the crosstalk and the PA output signals. Despite that, their approach can be applied to compensate for the combined effects of PA nonlinearity and antenna crosstalk in cases where the crosstalk can be considered small in power. Also, Amin et al. [15] use a system model without mixing of antenna crosstalk and PA output for their behavioral models and DPD of a two-path transmitter. Nevertheless, their DPD includes the necessary terms to compensate for the effects caused by any kind of antenna crosstalk. Another method that could be used to combat the nonlinear effects due to antenna crosstalk is presented in [16], where a method for joint nonlinearity and in-phase and quadrature imbalance in multiantenna transmitters is proposed. However, these techniques do not scale well for systems with a larger number of antennas, since they would require a multi-input DPD, i.e., a multivariate polynomial or Volterra series, in every path of the transmitter. Due to complexity, such an approach is not feasible for emerging transmitters with many antennas. To reduce the complexity of such approaches, Zenteno et al. [17] propose a sparse estimation technique that can reduce the complexity but still requires the use of multi-input memory polynomial models. In this paper, we propose a multi-antenna transmitter DPD technique that employs a completely different structure than existing techniques in order to reduce complexity. The basis for the proposed technique is a multi-antenna transmitter model first presented in [4], where dual-input PA models are combined with linear antenna array models. The proposed DPD technique consists of two main blocks: one linear block that models antenna crosstalk and mismatch and is shared by all paths of the transmitter, and a nonlinear dual-input DPD block in every transmit path, as shown in Fig. 2. Our solution is suitable for multi-antenna transmitters with any kind of crosstalk and mismatch at the PA outputs that can be described as a linear function of all transmit path outputs. The complexity of the antenna crosstalk and mismatch block increases linearly with the number of transmit paths, while the complexity of each dual-input DPD block is completely independent of the number of transmit paths. Hence, for transmitters with more than two paths, the complexity of the proposed DPD is lower compared to existing solutions. Just like existing solutions, the proposed DPD can be identified from measurements of the individual PA output signals using conventional linear least-squares estimation algorithms, and requires no other prior knowledge of the characteristics of the PAs or the antenna array. The proposed DPD technique is evaluated and compared to existing techniques in measurements of a four-path transmitter. To the best of author’s knowledge, this is the first time that measurement results of a transmitter with more than two paths are presented for these kinds of DPDs. This paper is organized as follows. In Section II, we introduce the system model of a multi-antenna transmitter.

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Fig. 2. Block diagram of a multi-antenna transmitter with the proposed DPD method. The method consists of two main blocks: one linear CTMM block for the whole transmitter and a dual-input DPD block in every transmit path.

In Section III, we present the proposed multi-antenna transmitter DPD including an identification procedure. The proposed technique is evaluated in measurements of a four-path transmitter. The setup for the experiments is explained in Section IV, and the results are presented in Section V. Finally, we draw our conclusions in Section VI. II. S YSTEM M ODEL In this section, the system model of the multi-antenna transmitter in equivalent discrete-time lowpass description is given. Note that, where applicable, the time dependence is omitted for better legibility, such that, for example, aik (n) is written as aik . A block diagram of a multi-antenna transmitter with K transmit paths is shown in Fig. 1. Each transmit path consists of an RF PA connected to an antenna element. The antenna elements form the antenna array. All transmit paths operate in the same frequency band. The signal a1k is the input to the PA of the kth path, and the signal b2k is the PA output signal. Due to antenna crosstalk and mismatch, a signal a2k is incident to the output of the PA. Each PA of the transmitter can be modeled as a dual-input system with one output. The crosstalk and mismatch signal a2k is a function of the PA output signals of all paths, and the relation between a2k and the output signals b2k is determined by the characteristics of the antenna array. The system model of the multi-antenna transmitter can, therefore, be split in two parts: a crosstalk and mismatch model (CTMM) and a dual-input PA model [4], [5]. A. CTMM The CTMM describes the crosstalk and mismatch signals a2k as a function of the PA outputs b2k . If the antennas are wideband compared to the signal bandwidth, a2k can be described as a linear combination of the PA output signals of all transmit paths by [4] and [5] a2k =

K  i=1

λki b2i = b2T λk

(1)

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where b2 = [b21 , . . . , b2K ]T , λki are complex coefficients, and λk = [λk1 , . . . , λk K ]T . The antenna array scattering parameters (S-parameters) can be used to describe the characteristics of an antenna array. The S-parameters measured at center frequency, i.e., the S-parameter matrix of the array, then correspond to a matrix [λ1 , . . . , λ K ]. B. Dual-Input PA Model The PA output signal b2k of the kth PA is modeled as a function of the signals a1k and a2k by [5] and [18] b2k =

( P−1)/2−1 p  p+1  

=

∗ p−v ∗ v θkpvu a1k p+1−u a1k a2k u a2k

∗ p αkp a1k p+1 a1k

p=0

+

( P−1)/2 

∗ p βkp a1k p a1k a2k

p=0

+

( P−1)/2 

∗ p−1 ∗ γkp a1k p+1 a1k a2k

p=1

+

( P−1)/2 p  p+1  

δkpuv v=0 u=0 u>1−v ∗ p−v a u a ∗ v a1k p+1−u a1k 2k 2k p=1

×

⎫ ⎬ ⎭ ⎫ ⎬

(2a)

⎭ ⎫ ⎬

(2b)

⎭ ⎫ ⎪ ⎪ ⎬

(2c)

⎪ ⎪ ⎭

(2d)

where θkpvu , αkp , βkp , γkp , and δkpuv are complex coefficients. As can be seen, there are four types of basis functions, which all contain one more nonconjugate term than conjugate terms [7]: basis functions that depend only on a1k in (2a), basis functions that depend on a1k and linear terms of a2k in (2b), ∗ basis functions that depend on a1k and linear terms of a2k in (2c), and basis functions that depend on a1k and nonlinear terms of a2k in (2d). If the crosstalk and mismatch signal a2k can be considered relatively small in power, only linear terms of a2k need to be considered in the dual-input PA model [19]. Then, all basis functions in (2d) become negligible and can be set to zero. To make the dual-input PA model suitable for wideband signals, dynamic effects need to be considered [6]. This is done by introducing memory effects. In [15] and [18], several strategies for introducing memory effects in dual-input PA models are explained. Any of these strategies can be used in the models presented here. While the concept of introducing memory is rather simple, the equations become cumbersome to read. We, therefore, describe the memoryless equations in the main text here and refer the reader to Appendix A, where we present the equations including memory effects. In matrix form, the dual-input PA model can be written as  b2k = G(0) (a1k ) G(1)(a1k , a2k ) G(2) (a1k , a2k )  T G(3)(a1k , a2k ) α kT β kT γ kT δ kT = G(a1k , a2k )θ k

where a1k , a2k , and b2k are the vectors containing all the time samples of the signals a1k , a2k , and b2k , e.g., a1k = [a1k (0), . . . , a1k (N − 1)]T , where N is the number of samples. The vectors α k , β k , γ k , δ k , and θ k contain the complex coefficients, e.g., α k = [αk0 , . . . , αk(( P−1)/2−1) ]T . Furthermore, the matrix G(0)(a1k ) contains the basis functions in (2a), G(1)(a1k , a2k ) contains the basis functions in (2b), G(2)(a1k , a2k ) contains the basis functions in (2c), G(3)(a1k , a2k ) contains the basis functions in (2d), and G(a1k , a2k ) combines all basis functions in (2). III. M ULTI -A NTENNA T RANSMITTER DPD

v=0 u=0

p=0 ( P−1)/2 

3

(3)

We propose a DPD that consists of two main blocks: one linear CTMM block for the whole transmitter, and a dualinput DPD block in every transmit path. A block diagram of the proposed method is shown in Fig. 2. Each dual-input DPD is the inverse function of the respective dual-input PA, while the CTMM block emulates the behavior of the antenna array. One input of the kth dual-input DPD is the signal bdk that is the desired output signal of the kth PA, i.e., in a perfectly linearized transmitter b2k = bdk . The second input to the kth dual-input DPD is an estimate aˆ 2k of the crosstalk and mismatch signal. The CTMM block creates the signals aˆ 2k from the signals bdk . The output of the kth dual-input DPD, which is driving the kth PA, is the signal a1k . The identification of the coefficients in the proposed technique is based on the measurements of the individual PA output signals, and requires no other prior knowledge of the PA behavior or the characteristics of the antenna array. First, using the signals bdk , which are known, the CTMM block coefficients can be identified from the measurements of the PA output signals b2k . Then, with the signals bdk and the output of the CTMM block aˆ 2k , the dual-input DPD coefficients can also be identified from the measurements of the PA output signals b2k . Note that for the identification of both the dual-input DPD and the CTMM coefficients, the input signals bdk to the different transmit paths cannot be fully correlated. For the special case of multi-antenna systems with fully correlated signals like, e.g., beamforming systems, suitable training signals have to be used for the identification, which requires the transmission of user data to be interrupted. However, for applying the CTMM and the DPD, no such restrictions apply. The CTMM block and dual-input DPD as well as the identification procedures are described below. Note that, without loss of generality, we assume amplifiers with unity gain to simplify the notation. When applying the proposed technique to PAs with nonunity gain, any conventional gain normalization concept can be used (see [20], [21]). A. CTMM The CTMM produces the signals aˆ 2k by aˆ 2k = bdT λˆ k

(4)

where λˆ k is a vector with the CTMM coefficients of the kth transmit path, which have to be identified. The vector bd = [bd1, . . . , bd K ]T contains the input signals bdk to the

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Fig. 4. Flowchart for the identification of the dual-input DPD coefficients of the kth transmit path. Fig. 3. Flowchart for the identification of the CTMM coefficients of the kth transmit path.

transmitter, which are known. Also known are the PA output signals b2k , which are obtained by measurements. Because no DPD is applied for the CTMM coefficient identification procedure, a1k = bdk . The CTMM coefficients λˆ k are identified for every transmit path individually in a two-step procedure. First, the coefficients αˆ k , βˆ k , γˆ k , and δˆ k of the dual-input PA model in (3) are estimated. Then, using the estimated PA model coefficients, the CTMM coefficients λˆ k are estimated. The two steps of the procedure are performed for several iterations until the result is satisfying. In the initial step, all CTMM coefficients λˆ k are arbitrarily set to 1. A simple flowchart of the procedure is shown in Fig. 3. In step 1 of the procedure, using (3), least-squares estimates αˆ k , βˆ k , γˆ k , and δˆ k of the PA model coefficients are obtained with   T T T αˆ kT βˆ k γˆ kT δˆ k = G(a1k , aˆ 2k )+ b2k (5) where the pseudoinverse G+ = (G H G)−1 G H is used. In step 2 of the procedure, which is fully derived in Appendix B, an estimate of the CTMM coefficients λˆ k = {λˆ k } + j {λˆ k } is found with

 (1) (2) (1) (2) + {Fk + Fk } {−Fk + Fk } {λˆ k } = (1) (2) (1) (2) {λˆ k } {Fk − Fk } {Fk + Fk }  (0) (3) {b2k − fk − fˆk } (6) × (0) (3) {b2k − fk − fˆk } where {·} and {·} denote the real and imaginary parts, respectively, and (1)

(1)

(2)

(2)

Fk = diag(fk )B2 Fk = diag(fk )B∗2

samples, are obtained from (0) fk

=

( P−1)/2 

∗ p αˆ kp a1k p+1 a1k

p=0 (1)

fk

=

( P−1)/2 

∗ p βˆkp a1k p a1k

p=0

f k(2) =

( P−1)/2 

∗ p−1 γˆkp a1k p+1 a1k .

(8)

p=1

Furthermore, (3)

f k (λˆ k ) =

( P−1)/2 p  p+1   v=0 u=0 u>1−v

p=1

∗ p−v δˆkpuv a1k p+1−u a1k

u v

ˆ∗ . b∗T × b2T λˆ k 2 λk

(9)

As can be seen, f k(3)(λˆ k ) is nonlinear in the crosstalk coefficients λˆ k . However, it is not desirable to solve this nonlinear problem due to computational requirements. There are two alternatives to avoid this. The first alternative is to set λˆ k = 0 (3) (3) (3) within f k , such that fˆk = f k (0) = 0. The second option is to use a previous estimate λˆ k . Note that for a dual-input PA model using only linear terms of a2k , f k(3) (λˆ k ) = 0 inherently. For a system model with memory effects, the equations for step 2 need be adapted. This is shown in Appendix C. Note that another option of obtaining a set of coefficients for the CTMM block is to use antenna S-parameter measurements. We will show the results for both S-parameter measurements and the identification procedure outlined above in the experiment section. B. Dual-Input DPD

(7)

where diag(f) denotes a diagonal matrix with the elements of the vector f as diagonal entries and B2 = [b21 , . . . , b2K ]. The (0) (1) (2) vectors fk , fk , and fk , which contain values for all time

Fig. 2 illustrates the proposed DPD method. The predistorted signal a1k , i.e., the output of the dual-input DPD of the kth transmit path, is calculated for all time samples by a1k = H(bdk , aˆ 2k )ϕ k

(10)

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5

Fig. 6. Measurement setup in the laboratory showing measurement instruments, PAs, and antenna array. PA1 is connected to antenna 1 of the array, PA2 to antenna 2, and so on.

Fig. 5.

Block diagram of the measurement setup.

where ϕ k are the dual-input DPD coefficients of the kth transmit path, which have to be identified. The signals bdk to the DPD are known. The second input signals aˆ 2k to the DPD are produced by the CTMM block, as discussed in Section III-A. The dual-input DPD matrix H(bdk , aˆ 2k ) contains basis functions of the same kind as the dual-input PA model matrix G(a1k , a2k ) in (3). The PA output signals b2k are obtained by measurements. Fig. 4 shows a flowchart for the identification of the dualinput DPD coefficients ϕ k . The least-squares estimate of the dual-input DPD coefficients ϕ k is found as +

ϕ k = H(b2k , aˆ 2k ) a1k .

(11)

In the initial step of the dual-input DPD identification, a1k = bdk . The dual-input DPD identification is iterated until the resulting linearity is satisfying. This procedure is exactly the same as a conventional indirect learning architecture [21], with the extension that there is a known second input to the DPD. The proposed dual-input DPD can be used as the regional model in a vector-switched DPD, which was proposed to improve the results of PA linearization techniques in [22]. In this approach, several switching regions are defined, and a separate DPD is calculated and applied for each region. The switching regions for the dual-input DPD are based on the signals bdk . C. Nonuniqueness of Coefficients The solution for CTMM coefficients λˆ k and dual-input DPD coefficients ϕ k is not unique. Multiplying the CTMM coefficients by an arbitrary scaling factor while reciprocally scaling dual-input DPD coefficients will result in the same

DPD output signals, and therefore not change linearization performance. There is also ambiguity in the kth element of each of the CTMM coefficient vector λˆ k , as can be seen from (17), where b2k appears on both sides of the equation. This can cause numerical problems or unidentifiability. One way to prevent either CTMM coefficients or dual-input DPD coefficients from becoming arbitrarily large or small, is to keep the kth element of λˆ k to a fixed value, and to normalize the other CTMM coefficients of the vector after each CTMM identification step 2 to a suitable value by dividing all of them by a scaling factor. For example, after each step 2 of the CTMM identification, perform the normalization λˆ k /ρk , where ρk = max j, j =k (λˆ k ( j )), and set λˆ k (k) = 1. By doing so, CTMM coefficients with absolute values between zero and one are obtained. This is the approach we have taken in our studies. IV. E XPERIMENTAL VALIDATION In this section, the measurement setup of a four-element transmitter is presented, which is used to validate the proposed DPD technique and compare it to existing techniques. Measurement settings and performance evaluation measures are given as well. The results are presented in Section V. A. Measurement Setup A block diagram of the measurement setup of the fourelement transmitter is shown in Fig. 5, and a photograph of the setup is shown in Fig. 6. The four driving signals for the PAs were created in MATLAB. The signals were different and independent orthogonal frequency-division multiplexing signals with 5-MHz bandwidth, and peak-to-average power ratios of around 8.5 dB. Two synchronized high-speed dual-channel arbitrary waveform generators (AWG, Agilent M8190A) were used to synthesize the four driving signals. The four PAs were identical GaAs PA evaluation boards from Skyworks (SKY66001-11), supplied with 3.3 V and operated at a center frequency of 2.12 GHz. The instantaneous gain of one of the PAs is shown in Fig 7. The PAs have integrated couplers at their outputs, which were used to measure the individual PA

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B. Evaluated DPD Techniques and Performance Measures

Fig. 7. PA gain versus output power for one of the PAs (PA1). The PA gain is shown for the PA when used in a single-path transmitter (yellow), and for the PA when used in the presented four-path transmitter (blue).

Fig. 8. Measured antenna array S-parameters versus frequency. The characteristics are only shown for antenna 1, since they are similar for all antennas due to reciprocity. The figure shows the scattering parameters for: reflection (S11), adjacent element (S14), opposite element (S12), and diagonally opposite element (S13). The antenna array can be seen in the measurements setup in Fig. 6.

output signals. The antenna array was a rectangular fourelement array with microstrip patch elements. The highest coupling factor between two elements of the array is around −12 dB. The measured array scattering parameters versus frequency are shown in Fig. 8 for one antenna in the array. The other S-parameters show similar behavior due to symmetry of the antenna array design, which can be seen in Fig. 6. A single antenna element of the same type as the transmitter elements was used as a receiver. The four individual PA output signals were measured with a four-channel 8-bit oscilloscope (Rohde&Schwarz RTO1044). The noise floor of the OSC was the limit for the achievable linearization. The received signal, which is in the far-field of the four-element transmitter, was measured with a vector signal analyzer (Agilent PXA N9030a). The received signal is a linear combination of the four transmitted signals including channel effects as well as the effect of the receiver antenna. Since independent signals are feeding the four antennas, the resulting relative distortion of the received signal becomes independent of direction. Hence, the nonlinear distortion observed at the receiver is independent of the receiver location, which is why measurement results are presented for only one position. Processing was done in MATLAB at a baseband sampling frequency of 25 MHz. Dynamic effects are characteristic for PAs in a wideband scenario. A driving signal bandwidth of 5 MHz was chosen since it was wide enough to cause dynamic effects in the chosen PAs, while the antenna array characteristics were still approximately the same for all frequencies within the signal bandwidth.

The proposed DPD technique was tested and compared to single-input DPD and multi-input DPD, as well as the case where no DPD is used. As mentioned, an alternative to identifying a set of CTMM coefficients from measurements of the PA output signals is to use measurements of the antenna array S-parameters at center frequency. Results of the proposed technique are presented for both these methods. A big disadvantage of using measured S-parameters as CTMM coefficients is that a separate calibrated setup is required to obtain the S-parameters. The S-parameter measurements were performed with a twoport vector network analyzer directly at the antenna ports. The measurements were performed pairwise, with the other ports terminated in 50 . Results of the proposed technique where the CTMM coefficients were identified from PA output measurements are indicated with DI-CTMM DPD, and results using S-parameters with DI-SParam DPD. All multi-antenna DPD results, i.e., the proposed technique as well as the multi-input DPDs, are shown for two system models. The first system model, which is indicated by NLCT, is a memory polynomial dual-input PA model that considers nonlinear terms of the crosstalk signals a2k as given in (15). The second system model, indicated by LCT, considers only linear terms of the crosstalk signals a2k such that all terms in (15d) become zero. The LCT model is suitable if the crosstalk signals a2k are relatively small in power. Both system models consider cross-products between PA output and antenna crosstalk signals. The advantage of the LCT model is that it leads to much lower complexity than the NLCT model. For the CTMM identification with NLCT, a previous estimate of the crosstalk coefficients λk was used in f (3) in (21). Note that the LCT multi-input DPD is based on the technique in [14], and the NLCT multi-input DPD is based on the technique in [15]. Both techniques have been extended toward a four-path transmitter. For all DPDs, a vector-switched memory-polynomial DPD structure [22] with four switching regions based on the desired output signals bdk was used. For the evaluation of results, the normalized mean square error (NMSE) and the adjacent channel leakage ratio (ACLR) are used. The NMSE is calculated by  N−1 2 |x(n) − x(n)| ˆ (12) NMSE = n=0  N−1 2 n=0 |x(n)| where N is the number of time samples, x(n) is the desired output signal, and x(n) ˆ is the measured output signal. The ACLR is calculated as   2 f (ad j )c |X ( f )|  (13) ACLR = max 2 c=1,2 f ch |X ( f )| where X ( f ) is the measured amplitude spectrum, f ch denotes inband frequencies, and f ad j frequencies in the adjacent channel. V. R ESULTS In this section, we first evaluate the CTMM identification procedure, since its reliability is integral for the proposed

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7

TABLE I ACLR AND NMSE FOR D IFFERENT DPD T ECHNIQUES

DPD technique. Then, the results for the proposed DPD technique are presented, evaluated, and compared to existing DPD techniques. A. Evaluation of CTMM Identification The CTMM coefficients are an essential part of the proposed DPD technique. Therefore, it is important that the identification procedure is reliable. The reliability of the CTMM identification method using PA output measurements as proposed in Section III-A is evaluated by extracting CTMM coefficients for different initial values and comparing the results. The initial values were complex numbers with real and imaginary parts that were randomly chosen according to a uniform distribution with interval [−1, 1]. We have used the procedure suggested in Section III-C to avoid numerical and identification problems. Using this procedure, the kth CTMM coefficient and the maximum CTMM coefficient of the kth path assume the fixed value 1. The results for identification of the remaining CTMM coefficients for all paths are shown in Fig. 9(a) for LCT and in Fig. 9(b) for NLCT. Each blue dot represents an initial value, and the yellow circles show the results after the first iteration. The black plus signs show the results after the second iteration. As can be seen, the results converge to the same value for all initial values. After the second iteration, no difference between the results for different initial values can be noted. Furthermore, the final results for LCT and NLCT agree. The accuracy of the identified CTMM coefficients becomes evident from the performance evaluation of the proposed DPD technique, which is presented in the following section. B. Performance of the DPDs In Table I, the ACLRs for the received signal and each of the individual PA output signals are given for different DPD methods. For the case without any DPD, the ACLR is between −40 and −36 dB. Using single-input DPD improves the ACLR to around −45 dB. The ACLR results for the proposed DI-CTMM and DI-Sparam DPDs and the multi-input DPDs are very similar, around −50 dB, for both NLCT and LCT. The multi-input LCT DPD reaches the best result. As can be seen,

Fig. 9. Results of the CTMM coefficient identification based on (a) LCT and (b) NLCT. The figures show the initial values in blue dots, the results after the first iteration in yellow circles, and the results after the second iteration in black plus signs. As can be seen, the results converge such that after the second iteration no difference can be seen. The results for LCT and NLCT agree.

linearizing each transmit path via the measured PA output signals improves the ACLR of the received signal to a similar degree as the ACLR of the individual PA output signals. The NMSEs for different DPD techniques are given in Table I. Results are shown for each of the individual PA output signals. Since we have no knowledge of the channel, it is not relevant to evaluate the NMSE for the received signal. Compared to the case without any DPD, the single-input DPD does not significantly improve the NMSE. The proposed DI-SParam DPD achieves an improvement of around 12 dB for both NLCT and LCT. The proposed DI-CTMM DPD and the multi-input DPD improve the NMSE by around 19 dB for both NLCT and LCT. The NMSEs for the proposed

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Fig. 10. Spectra of the PA output signals. The figure shows the results without DPD (blue ), with single-input DPD (SI DPD, red ), multi-input DPD (MI LCT DPD, purple ◦), and the proposed DPD (DI-CTMM LCT, yellow ♦). The multi-input DPD and the proposed DPD are based on models using only linear terms of the crosstalk signals.

C. Complexity The issue of complexity has many aspects. There are different types of complexity that should be considered [23].

Fig. 11. Normalized spectra of the received signals. The figure shows the results without DPD (blue ), with single-input DPD (SI DPD, red ), multiinput DPD (MI LCT DPD, purple ◦), and the proposed DPD (DI-CTMM LCT, yellow ♦). The multi-input DPD and the proposed DPD are based on models using only linear terms of the crosstalk signals.

DI-CTMM DPDs and the multi-input DPDs are very similar, with the multi-input LCT DPD achieving the lowest values. The lowest values for ACLR and NMSE are achieved by the multi-input LCT DPD and the proposed DI-CTMM LCT DPD. In Fig. 10, the spectra of the individual PA outputs are shown, and in Fig. 11, the normalized spectra of the received signals are shown for these two DPD techniques. Also shown for comparison are the results for single-input DPD and for the case without any DPD. As can be seen, the single-input DPD reduces the out-of-band distortion. However, both DICTMM LCT DPD and multi-input LCT DPD reach a much better result. The results for the received signal, which is the far-field of the transmitter, are equal to the results of the PA output signals. Note that the performance of all linearization techniques is limited by the noise floor of the OSC, which was used to measure the PA output signals. The noisefloor is indicated in the figures.

1) Run-time complexity is the complexity to execute the DPD on the input signals. It depends on the number of calculations that are necessary for each input signal sample to obtain the predistorted signal. Therefore, runtime complexity also depends on signal and evaluation bandwidth, i.e., the required sampling rate of the system. 2) Identification complexity is the complexity required to find the initial version of the predistorter. This is typically done using least-squares techniques [29]. Identification complexity depends on the number of parameters that needs to be identified. The initial identification is only performed once, usually in the lab or factory. After that, adaptation is used to adjust the predistorter to changes in the system. For this reason, identification complexity is negligible and the focus is put on adaptation complexity. 3) Adaptation complexity is the complexity to adjust the identified predistorter to changes in system behavior while the system is running. Adaptation is commonly done using algorithms like least mean squares, recursive least squares, or similar [10]. Using these techniques, every coefficient of the predistorter is updated individually at run-time. Adaptation complexity depends on how much and how fast the systems change over time due to, e.g., temperature drift. The exact complexity and related measures, such as power consumption, cost, and space, always depend on a specific implementation, i.e., implementation concept [10], [24], [25], used hardware, necessity and frequency of adaptation, training algorithm, adaptation algorithm, matrix inversion algorithm, and bandwidth requirements.

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9

TABLE II N UMBER OF DPD C OEFFICIENTS PER T RANSMIT PATH PER S WITCHING R EGION AND THE N UMBER OF CTMM C OEFFICIENTS PER PATH R EQUIRED FOR THE D IFFERENT DPD T ECHNIQUES

We have chosen to use the number of predistorter coefficients as a basis for our complexity analysis. Comparing the number of predistorter coefficients is a simple way of comparing the complexity of the different DPDs. It can also be used to investigate how complexity scales when increasing the numbers of transmit paths, which serves as an indicator of whether a DPD technique is feasible for larger multi-antenna systems or not. For predistorters based on the Volterra series, as the ones considered in this paper, all types of complexity depend on the number of predistorter coefficients [26]. Therefore, when considering such DPD structures with similar types of basis functions and the same requirements for evaluation bandwidth, complexity comparison is commonly based on comparing the number of DPD coefficients. Thus, reducing the number of coefficients is often used as a technique to reduce complexity [14], [17], [26]–[28]. For single-input DPD and the multi-input DPDs, all types of complexity scale with the number of DPD coefficients. For the proposed DPDs, run-time and adaptation complexity scale with the total number of predistorter coefficients, which includes the dual-input DPD coefficients and the CTMM coefficients. Table II compares the number of DPD coefficients per transmit path per switching region and the number of CTMM coefficients. Note that these numbers are specific to the multiantenna transmitter that is used in the measurements. The proposed DI-CTMM and DI-SParam DPDs require a fewer coefficients than the multi-input DPDs. The number of coefficients required for multi-input NLCT DPD is excessively higher than for the other DPDs. Except for the single-input DPD, the lowest number of coefficients is required for the proposed DPD based on LCT. The table also shows how the number of coefficients would scale for a K -path transmitter with the same type of components. This is also illustrated in Fig. 12, where the number of DPD coefficients per switching region plus the number of CTMM coefficients are plotted versus the number of transmit paths K . It is important to realize that for all multiinput DPDs the number of coefficients increases rapidly with the number of paths, while for the proposed DPDs only the number of CTMM coefficients increases and the dual-input DPD is not affected. For multi-antenna systems with many

Fig. 12. Illustration of how the number of coefficients per path would scale with the number of transmit paths K . The figure shows the number of DPD coefficients per switching region plus the number of CTMM coefficients. Numbers are shown for: single-input DPD (SI DPD), the proposed DPD (proposed LCT), and multi-input DPD (MI LCT DPD) based on models with only linear crosstalk terms, the proposed DPD (proposed NLCT), and multiinput DPD (MI NLCT DPD) based on models using linear and nonlinear terms of the crosstalk signals.

antennas, the difference in total coefficient numbers becomes huge, as can be seen in Fig. 12. The major advantage of the proposed technique is how it scales for the increasing numbers of transmit paths. The numbers given in Table II and Fig. 12 are specific to the multi-antenna transmitter used in the measurements and can, therefore, not give an exact prediction of the number of coefficients for other systems. However, due to its structure, the proposed technique will inherently have lower run-time complexity than the existing approaches for any transmitter, where antenna crosstalk from more than one transmit path needs to be considered in the DPD. D. Discussion Analyzing the presented results, several things can be noted. It is obvious that single-input DPD is not suitable for a multiantenna transmitter. While the ACLR can be improved with such a DPD, the NMSE is almost the same as when using no DPD at all. The ACLR results for the proposed DI-CTMM and DI-SParam DPDs are very similar. However, the NMSE results are worse for the DI-Sparam DPD. The reason is that the reference plane for the measurements of the S-parameters is not exactly the same as the reference plane for measurements of the PA output signals. Observations at different reference planes can have different phase shifts, gain shifts, and delays. Such issues can lead to a degradation of performance, or even failure of the DPD to linearize the system. With careful calibration, the phase, gain, and delay offsets were kept very small, such that a performance degradation when using DI-SParam DPDs is only noticeable in terms of NMSE while the ACLR stays unaffected. The CTMM coefficients estimated directly from PA output signals allow for better DPD results since they are obtained from measurements in the exact reference plane. In addition to the worse NMSE results and the complicated calibration, another disadvantage of using S-parameters as CTMM coefficients is that a separate measurement setup is required. In highly integrated transmit-

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ters, these measurements might not be feasible. Furthermore, for a system implementation, it is not possible to use adaptive algorithms to update the CTMM when using the S-parameters instead of the proposed identification method. All results obtained with the proposed DI-CTMM DPD are almost the same as the results of the multi-input DPDs. For both these techniques, the DPDs based on NLCT have much higher complexity than the DPDs based on LCT. Therefore, for the multi-antenna transmitter used in these measurements, there is no advantage in using the NLCT-based DPDs. Hence, for a maximum coupling between the antennas of −12 dB, it is not necessary to consider a dual-input PA model with nonlinear crosstalk and mismatch terms. However, for systems with higher coupling, it can be expected that NLCT-based DPDs are more suitable [18]. Considering both NMSE, ACLR, and the complexity, the best results are obtained by the proposed DI-CTMM LCT DPD. Even though the DI-CTMM LCT DPD is narrowly outperformed by the multi-input LCT DPD, the number of DPD coefficients is reduced. For emerging multi-antenna systems with the large numbers of transmit paths, the complexity of the proposed technique scales much better than for existing solutions, as shown in Table II and Fig. 12. The presented results also indicate that in cases where the coupling is so high that NLCT -based DPDs are required, existing solutions will quickly reach the limits of what is feasible to implement, while the complexity of the proposed technique still rises only relatively slowly with the number of transmit paths. Hence, the proposed DPD technique is an extremely attractive alternative for the linearization of emerging multi-antenna transmitters. VI. C ONCLUSION In this paper, we present a technique to jointly compensate for the effects of PA nonlinearity, antenna crosstalk, and mismatch in wideband multi-antenna transmitters. Dual-input DPDs in every transmit path are combined with a linear model of the crosstalk and mismatch characteristics of the antenna array that is shared by all transmit paths. Using this structure, the use of multi-input DPDs is avoided. Therefore, the complexity of the proposed technique scales very favorably with the number of transmit paths: the CTMM block increases linearly with the number of paths, while the complexity of the dual-input DPDs is not affected by the number of paths. This is a huge benefit of our technique compared to existing approaches, especially considering the trend toward large-scale multi-antenna systems. As we have shown, no prior knowledge of the system components is required to identify the dual-input DPDs or the CTMM. Similar to conventional DPD approaches, all necessary information can be identified from measurements of the PA output signals using least-squares estimation techniques. A potential disadvantage of the proposed technique is that, for the identification procedure, the input signals bdk to the different transmit paths cannot be fully correlated, such that for systems with fully correlated signals like, e.g., beamforming systems, special training signals have to be used.

Results are shown for the linearization of a four-path transmitter. When comparing the proposed technique to existing approaches, it can be seen that the performances are similar, while our technique has lower complexity. With the presented measurements, we show that our technique is suitable for the linearization of wideband PAs. A topic for future work is also to include dynamic antenna behavior in our algorithms. This can be necessary for systems where the antenna characteristics vary strongly within the input signal bandwidth. A PPENDIX A. Dual-Input PA Models Including Memory Effects To account for dynamic effects in PAs driven by wideband signals, memory has to be considered in the dual-input PA model. The most general form of a dual-input PA model with memory effects is a dual-input model according to the Volterra series [30] b2k (n) =

M 1  

 1−q1 θk0q1 0m 1 a1k (n − m 1 )

q1 =0 m 1 =0

 q × a2k (n − m 1 ) 1 +

p+1 ( P−1)/2   p=1 M 

···

m 2 p+2−q2 =0

×

M 

M 

···

m 2 p+1−q2 =m 2 p−q2

θkpq1 q2 m 1 m 2 ...m 2 p+1

m 2 p+1 =m 2 p p+1 

a1k (n − m i )

i=1 2 p+1−q  2 s= p+2

m p+1−q1 =m p−q1

M 

m p+1 =m p m p+2 =0

M 

p+1−q  1

M 

···

M 

···

q1 =0 q2 =0 m 1 =0

m p+2+q1 =0

×

p  M 

a2k (n − m l )

l= p+2−q1 2 p+1

∗ a1k (n − m s )

∗ a2k (n − m r ).

r=2 p+2−q2

(14) Because of the high model complexity, the full Volterra series approach is usually infeasible. Therefore, many models with reduced complexity have been proposed, such as the memory polynomial [6] and the generalized memory polynomial [7]. For the evaluation of the proposed DPD in measurements, we used the memory polynomial approach. The dual-input memory polynomial PA model is given in the same structure as (2) by ⎫ M ( P−1)/2 ⎪   ⎪ ⎬ αkpm 1 a1k (n − m 1 ) b2k (n) = (15a) m 1 =0 p=0 ⎪ ⎪  2 p ⎭ ×a1k (n − m 1 ) ⎫ M  ⎪ ⎪ ⎪ + βk0m 1 a2k (n − m 2 ) ⎪ ⎪ ⎪ ⎪ m 2 =0 ⎬ ( P−1)/2 M M    (15b) ⎪ + βkpm 1 m 2 ⎪ ⎪ ⎪ ⎪ ⎪ m 1 =0 m 2 =0 p=1 ⎪  2 p ⎭ ×a2k (n − m 2 )a1k (n − m 1 )

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⎫ ⎪ ⎪ ∗ + γkpm 1 m 2 a2k (n − m 2 )⎬ m =0 m 2 =0 p=1 ⎪ 1  p+1  ∗  p−1 ⎪ ⎭ × a1k (n − m 1 ) a1k (n − m 1 ) ⎫ p  p+1 M ( P−1)/2 M  ⎪    ⎪ ⎪ + δkpuvm 1 m 2 ⎪ ⎪ ⎬ M M  

( P−1)/2 

m 1 =0 m 2 =0

p=1

v=0 u=0

  p+1−u  u>1−v  ⎪ ∗ (n − m ) p−v ⎪ ⎪ a1k ×a1k (n − m 1 )  ⎪ 1 ⎪ v ⎭ u ∗ × a2k (n − m 2 ) a2k (n − m 2 ) .

(15c)

with B2 (n − m) = [b21 (n − m), . . . , b2K (n − m)] where b2k (n − m) = [b2k (0 − m), . . . , b2k (N − 1 − m)]T and

(15d)

m 1 =0 ( P−1)/2 M  

(2)

f km =

m 1 =0 p=1  ∗ × a1k (n −

fk

B. Derivation of CTMM Coefficient Identification

=

To find the equations for step 2 of the CTMM coefficient identification, (1) is introduced into (2)

+

( P−1)/2 

( P−1)/2 

∗

∗ p−1 ∗T ˆ γkp a1k p+1 a1k b2 λk

p=1

+

( P−1)/2 p  p+1   p=1

b2T λˆ k

u

v ˆ∗ b∗T 2 λk

(1)

(2)

(16)

∗

(3)

ˆ ˆ + f k b2T λˆ k + f k b∗T 2 λk + f k (λk ).

(17)

Using (7), this is rearranged and expressed for all time samples in matrix form as (2) ∗ b2k − fk(0) − fˆk(3) = F(1) k λk + Fk λk (3)

(18)

(3)

where fˆk is fk (λˆ k ) with λˆ k either chosen equal to zero or set to a value from a previous identification step as explained in Section III-A. Finally, by splitting it into real and imaginary parts, (18) can be solved for the CTMM coefficients as given in (6). C. CTMM Identification for Dual-Input Memory Polynomial PA Model Using the dual-input memory polynomial PA model for the identification of the CTMM coefficients, the equation given in (6) can be applied with the following adaptations: (7) changes to (1)

M 

(2)

m=0 M 

Fk = Fk =

m=0

(1)

diag(fkm )B2 (n − m) (2)

 p−1

.

(20)

M 

( P−1)/2 

m 1 =0

p=0

αˆ kpm 1 a1k (n − m 1 )

M M  

( P−1)/2 p  p+1   p=1

δˆkpuvm 1 m 2

v=0 u=0 u>1−v

ACKNOWLEDGMENT

which is then rewritten as (0)

m 1)

where b2 (n − m) = [b21(n − m), . . . , b2k (n − m)]T .

δkpuv

v=0 u=0 u>1−v

∗ p−v × a1k p+1−u a1k

b2k = f k

  p+1 γˆkpm 1 m a1k (n − m 1 )

  p+1−u  ∗  p−v a1k (n − m 1 ) × a1k (n − m 1 )

 T u × b2 (n − m 2 ) λˆ k

 T ∗ v × b∗2 (n − m 2 ) λˆ k (21)

∗ p Tˆ βkp a1k p a1k b2 λk

p=0

+

p=1

m 1 =0 m 2 =0

p=0

2 p  βˆkpm 1 m a1k (n − m 1 )

2 p  × a1k (n − m 1 ) f k(3) (λˆ k ) =

∗ p αkp a1k p+1 a1k

( P−1)/2 

Furthermore, (0)

( P−1)/2 

M 

(1) f km = βˆk0m +

The dual-input memory polynomial PA model can be written in matrix form exactly as (3).

b2k =

11

diag(fkm )B∗2 (n − m)

(19)

The authors would like to thank Skyworks Solutions, Inc. for donating the power amplifier test boards used in the experiments. R EFERENCES [1] “5G radio access,” Ericsson AB, Stockholm, Sweden, White Paper Uen 284 23-3204 Rev C, Apr. 2016, accessed: Mar. 2, 2017. [Online]. Available: https://www.ericsson.com/res/docs/whitepapers/wp5g.pdf [2] F. Rusek et al., “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013. [3] M. Romier, A. Barka, H. Aubert, J. P. Martinaud, and M. Soiron, “Load-pull effect on radiation characteristics of active antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 550–552, 2008. [4] C. Fager, X. Bland, K. Hausmair, J. C. Cahuana, and T. Eriksson, “Prediction of smart antenna transmitter characteristics using a new behavioral modeling approach,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2014, pp. 1–4. [5] K. Hausmair et al., “Prediction of nonlinear distortion in wideband active antenna arrays,” IEEE Trans. Microw. Theory Techn., to be published, doi: 10.1109/TMTT.2017.2699962. [6] J. Kim and K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory,” Electron. Lett., vol. 37, no. 23, pp. 1417–1418, Nov. 2001. [7] D. R. Morgan, Z. Ma, J. Kim, M. G. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3852–3860, Oct. 2006. [8] P. Suryasarman, M. Hoflehner, and A. Springer, “Digital pre-distortion for multiple antenna transmitters,” in Proc. Eur. Microw. Conf., Oct. 2013, pp. 412–415. [9] P. Suryasarman and A. Springer, “Adaptive digital pre-distortion for multiple antenna transmitters,” in Proc. IEEE Global Conf. Signal Inf. Process. (GlobalSIP), Dec. 2013, pp. 1146–1149.

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[10] P. M. Suryasarman and A. Springer, “A comparative analysis of adaptive digital predistortion algorithms for multiple antenna transmitters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 5, pp. 1412–1420, May 2015. [11] Z. Zhang, Y. Shen, S. Shao, W. Pan, and Y. Tang, “An improved cross talk cancelling digital predistortion for MIMO transmitters,” Mobile Inf. Syst., vol. 2016, Mar. 2016, Art. no. 5626495. [12] S. A. Bassam, M. Helaoui, and F. M. Ghannouchi, “Crossover digital predistorter for the compensation of crosstalk and nonlinearity in MIMO transmitters,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 5, pp. 1119–1128, May 2009. [13] M. V. Amiri, S. A. Bassam, M. Helaoui, and F. M. Ghannouchi, “Matrixbased orthogonal polynomials for MIMO transmitter linearization,” in Proc. IEEE Int. Workshop Comput. Aided Modeling, Anal. Design Commun. Links Netw., Dec. 2010, pp. 57–60. [14] A. Abdelhafiz, L. Behjat, F. M. Ghannouchi, M. Helaoui, and O. Hammi, “A high-performance complexity reduced behavioral model and digital predistorter for MIMO systems with crosstalk,” IEEE Trans. Commun., vol. 64, no. 5, pp. 1996–2004, May 2016. [15] S. Amin, P. N. Landin, P. Händel, and D. Rönnow, “Behavioral modeling and linearization of crosstalk and memory effects in RF MIMO transmitters,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 810–823, Apr. 2014. [16] Z. A. Khan, E. Zenteno, P. Händel, and M. Isaksson, “Digital predistortion for joint mitigation of I/Q imbalance and MIMO power amplifier distortion,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 1, pp. 322–333, Jan. 2017. [17] E. Zenteno, S. Amin, M. Isaksson, D. Rönnow, and P. Händel, “Combating the dimensionality of nonlinear MIMO amplifier predistortion by basis pursuit,” in Proc. Eur. Microw. Conf., Oct. 2014, pp. 833–836. [18] H. Zargar, A. Banai, and J. C. Pedro, “A new double input-double output complex envelope amplifier behavioral model taking into account source and load mismatch effects,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 2, pp. 766–774, Feb. 2015. [19] G. Z. El Nashef et al., “Second order extension of power amplifiers behavioral models for accuracy improvements,” in Proc. Eur. Microw. Conf., Sep. 2010, pp. 1030–1033. [20] A. Zhu, P. J. Draxler, J. J. Yan, T. J. Brazil, D. F. Kimball, and P. M. Asbeck, “Open-loop digital predistorter for RF power amplifiers using dynamic deviation reduction-based volterra series,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 7, pp. 1524–1534, Jul. 2008. [21] J. Chani-Cahuana, C. Fager, and T. Eriksson, “A new variant of the indirect learning architecture for the linearization of power amplifiers,” in Proc. Eur. Microw. Conf., Sep. 2015, pp. 1295–1298. [22] S. Afsardoost, T. Eriksson, and C. Fager, “Digital predistortion using a vector-switched model,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 4, pp. 1166–1174, Apr. 2012. [23] A. S. Tehrani, H. Cao, T. Eriksson, M. Isaksson, and C. Fager, “A comparative analysis of the complexity/accuracy tradeoff in power amplifier behavioral models,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 6, pp. 1510–1520, Jun. 2010. [24] A. Zhu, J. C. Pedro, and T. J. Brazil, “Dynamic deviation reductionbased Volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4323–4332, Dec. 2006. [25] Y. Ma, Y. Yamao, Y. Akaiwa, and C. Yu, “FPGA implementation of adaptive digital predistorter with fast convergence rate and low complexity for multi-channel transmitters,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 11, pp. 3961–3973, Nov. 2013. [26] F. M. Ghannouchi and O. Hammi, “Behavioral modeling and predistortion,” IEEE Microw. Mag., vol. 10, no. 7, pp. 52–64, Dec. 2009. [27] N. Kelly and A. Zhu, “Low-complexity stochastic optimization-based model extraction for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 5, pp. 1373–1382, May 2016. [28] Z. Wang, W. Chen, G. Su, F. M. Ghannouchi, Z. Feng, and Y. Liu, “Low computational complexity digital predistortion based on direct learning with covariance matrix,” IEEE Trans. Microw. Theory Techn., to be published, doi: 10.1109/TMTT.2017.2690290. [29] L. Guan and A. Zhu, “Optimized low-complexity implementation of least squares based model extraction for digital predistortion of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 3, pp. 594–603, Mar. 2012. [30] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, 2nd ed. Melbourne, FL, USA: Krieger, 2006.

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Katharina Hausmair received the Dipl.-Ing. degree in electrical and information engineering from the Graz University of Technology, Graz, Austria, in 2010, and is currently pursuing the Ph.D. degree at the Department of Electrical Engineering, Chalmers University of Technology, Göteborg, Sweden. From 2010 to 2013, she was a Researcher with the Signal Processing and Speech Communication Laboratory, Graz University of Technology. Her current research interests include signal processing for communication systems with an emphasis on modeling and compensation of undesired effects occurring in analog circuits. Per N. Landin received the M.Sc. degree from Uppsala University, Uppsala, Sweden, in 2007, and the Ph.D. degree from the KTH Royal Institute of Technology, Stockholm, Sweden, and Vrije Universiteit Brussel, Brussels, Belgium, in 2012. From 2013 to 2014, he was a Post-Doctoral Researcher with the Chalmers University of Technology, Göteborg, Sweden. Since 2015, he has been with Ericsson AB, Kumla, Sweden. His current research interests include RF measurements and signal processing, over-the-air measurements, and system identification applied to power amplifier modeling and linearization. Ulf Gustavsson received the M.Sc. degree in electrical engineering from Örebro University, Örebro, Sweden, in 2006, and the Ph.D. degree from the Chalmers University of Technology, Göteborg, Sweden, in 2011. He is currently a Senior Researcher with Ericsson AB Research, Göteborg, where he is also the Lead Scientist within the Marie Skłodowska-Curie Innovative Training Network, SILIKA. His current research interests include radio signal processing and behavioral modeling of radio hardware for advanced antenna systems. Christian Fager received the Ph.D. degree from the Chalmers University of Technology, Göteborg, Sweden, in 2003. Since 2015, he has been a Professor with the Microwave Electronics Laboratory, Chalmers University of Technology, where he is also the Deputy Director of the GHz Center for industrial collaborations. He has authored or co-authored more than 120 publications in international journals and conferences. His current research interests include energy efficient and linear transmitters for future wireless communication systems. Dr. Fager serves as a TPC member of the IEEE IMS and INMMiC technical conferences and is currently an Associate Editor for the IEEE Microwave Magazine. He was a recipient of the Best Student Paper Award of the IEEE MTT-S International Microwave Symposium in 2003. Thomas Eriksson received the Ph.D. degree in information theory from the Chalmers University of Technology, Göteborg, Sweden, in 1996. From 1990 to 1996, he was with the Chalmers University of Technology. In 1997 and 1998, he was with the AT&T Labs–Research, Murray Hill, NJ, USA. In 1998 and 1999, he was with Ericsson Radio Systems AB, Kista, Sweden. He was a Guest Professor with Yonsei University, Seoul, South Korea, from 2003 to 2004. Since 1999, he has been with the Chalmers University of Technology, where he is currently a Professor of communication systems, leading research on hardware-constrained communications. He is also the Vice Head of the Department of Electrical Engineering, Chalmers University of Technology, where he is responsible for undergraduate and master’s education. He is leading several projects on massive multi-in multiout (MIMO) communications with imperfect hardware, MIMO communication taken to its limits: 100-Gbit/s link demonstration, massive MIMO testbed design, satellite communication with phase noise limitations, and efficient and linear transceivers. He has authored or co-authored more than 200 journal and conference papers and holds 11 patents. His current research interests include communication, data compression, and modeling and compensation of nonideal hardware components (e.g., amplifiers, oscillators, and modulators in communication transmitters and receivers, including massive MIMO).

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Exposure Evaluation of an Actual Wireless Power Transfer System for an Electric Vehicle With Near-Field Measurement Jerdvisanop Chakarothai, Member, IEEE, Kanako Wake, Member, IEEE, Takuji Arima, Member, IEEE, Soichi Watanabe, Member, IEEE, and Toru Uno, Senior Member, IEEE

Abstract— In this paper, we propose an experimental approach for determining the internal electric field for exposure evaluation of wireless power transfer (WPT) systems by using measured magnetic near-field data. Two WPT systems are fabricated and used in the measurements: one without ferrite tiles, and the other with ferrite tiles and a metal plate. The amplitude and phase of the magnetic near field in the vicinity of the WPT systems are then measured by using in-house magnetic-field probes and a near-field measurement system. Numerical dosimetry of human exposure is performed using the measured near field as an incident field in the impedance method to derive the internal electric field strength inside numerical human models. Validation of the proposed approach has been demonstrated by comparing measurement results with those obtained from numerical simulations. Additionally, the coupling factor, which represents the relationship between the incident magnetic field and the induced electric field in the human body, at different distances is derived for realistic exposure scenarios. Index Terms— Coupling factor, electric vehicles, exposure assessment, magnetic near-field measurement, wireless power transfer (WPT) system.

I. I NTRODUCTION

R

ECENTLY, wireless power transfer (WPT) has been considered one of the emerging technologies and its application is rapidly increasing day by day [1]. The WPT technology provides convenience in charging many wireless devices used in daily life, such as mobile phones, notebook computers, and electric vehicles (EVs) [2]. Various types of inductive coupling WPT systems for transport systems such as an electric bus have been demonstrated [3]–[5]. There are many advantages of the wireless charging of EVs, e.g., no requirement of plugging in an EV with a power transmission wire and reduction of the possibility of leakage of electricity from a wet plug when it is rainy. A large electric power

Manuscript received February 20, 2017; revised June 9, 2017 and August 1, 2017; accepted August 15, 2017. This work was supported by the committee to promote research on the potential biological effects of electromagnetic fields, Ministry of Internal Affairs and Communications, Japan. (Corresponding author: Jerdvisanop Chakarothai.) J. Chakarothai, K. Wake, and S. Watanabe are with the National Institute of Information and Communications Technology, Tokyo 184-8795, Japan (e-mail: [email protected]). T. Arima and T. Uno are with the Department of Electrical and Electronics Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2748949

of about several kilowatts is, however, applied in the charging system of EVs. Consequently, a strong electromagnetic (EM) near field is produced around the WPT system during its charge operation. It is, therefore, necessary to evaluate human exposure to EM near-fields radiated from WPT systems. For a WPT system without countermeasures for the leakage of EM fields, the field strength may exceed limit levels prescribed in regulations. Specifically, compliance with human exposure guidelines is required for the development of WPT systems. Meanwhile, concerning the biological hazard of humanbody exposure to EM fields, international guidelines have been published by the International Commission of Non-Ionizing Radiation Protection (ICNIRP) [6], [7] and the IEEE [8], [9]. In these guidelines, there are two major biological effects needed to be considered when a human body is exposed to EM fields: thermal and stimulus effects. The exposure limit depends on the frequency. In the guidelines, the thermal effect is defined and limited in terms of the specific absorption rate (SAR) for frequencies higher than 100 kHz, while the stimulus effect is defined in terms of the internal electric field strength for frequencies lower than 10 MHz for ICNIRP guidelines and lower than 5 MHz for the IEEE. Since the SAR and internal electric field inside a human body are difficult to measure, the ICNIRP guidelines also provide reference levels, which are defined in terms of incident electric and magnetic-field strengths for the compliance test of wireless devices. The IEEE has also set similar guideline indexes in terms of incident electric and magnetic fields. The reference levels were determined by assuming a “worstcase” scenario of a human body in a standing position exposed to an EM plane wave. On the other hand, for the situation that the EM fields are highly nonuniform and the exposure is localized in a small region, such as exposure close to the WPT system, the coupling between the human body and the near field of the WPT system is less strong than that considered to derive the reference level. Consequently, the compliance test using the reference levels usually gives a very restricted limitation for WPT systems compared with that when the basic restrictions are applied as a metric (see [10]–[12]). The compliance with the basic restrictions is often done by numerical approaches since the measurements of the SAR and internal electric field inside biological bodies are difficult [10]–[12]. However, numerical approaches require high computer programming skills and/or high computational

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resources, which are in practice not available for device or product manufacturers. To reduce the complexity and requirements for the exposure evaluation, many attempts have been made to generalize induced quantities inside a human body, which are caused by incident EM fields, in a frequency regime where a magnetoquasistatic (MQS) approximation is applicable [13]–[15]. In addition, although numerical approaches can be used to determine the SAR and internal electric field inside a human body for compliance, it is difficult to accurately model an actual exposure situation in numerical simulations. For example, in some cases, we also need to model a vehicle body, vehicle tires, ground planes, and so forth. In addition, it is sometimes difficult to retrieve information about the size, shape, and design of WPT systems from manufacturers. In order to find the exposure level in a human model for a real situation, Wake et al. [16] proposed an experimental technique combining measured magnetic near-field and numerical analyses for induction-heating (IH) hobs. Although a similar concept is applied in this paper, the target frequency and magnetic-field distribution from WPT systems are considerably different from those of IH hobs since WPT systems are composed of transmitting and receiving parts. Therefore, the characteristics of the exposure are also different. Thus, we use a different system and magnetic-field probes to measure the magnetic near field for WPT systems operating at intermediate frequencies. The frequency range of the measurement was from 2 to 400 kHz for the previous system, while it is from 20 kHz to 10 MHz for our proposed system. In addition, there was no detailed information about phase measurement in [16]. In this paper, we propose an approach combining measurement and numerical analyses for evaluating localized EM exposure due to kilohertz-band WPT systems for EVs. Our proposed method uses both the magnitude and phase of the magnetic near field in numerical simulations to determine the exposure level in a realistic human model. The advantage of the proposed approach is that there is no need to know the shape or design of the WPT systems so that the approach is applicable to actual products of WPT systems. The paper outline is as follows. First, a measurement system for the magnetic near-field and WPT systems fabricated for measurements are described in Section II. Numerical human models, the numerical method, and exposure scenarios are described in Section III. Then, the WPT source used in numerical models is validated by comparison with the measurement results. Numerical simulations of exposure using the measured field data are performed to derive the internal electric field in human-body models, and the coupling factor is determined for the WPT system, as described in Section IV. Finally, conclusions are drawn in Section V. II. N EAR -F IELD M EASUREMENT S YSTEM AND WPT S YSTEMS A. Magnetic Near-Field Measurement System A measurement system was constructed and used to measure the magnetic near field of the WPT systems, as shown

Fig. 1. Measurement system for magnetic near field of WPT systems; SG: signal generator and PM: power meter. The distance between the transmitting and receiving coils of the systems is indicated by the symbol d.

Fig. 2. Schematic view and picture of fabricated magnetic-field probes. The size of the rectangular loops for detecting the magnetic field is 1 × 1 × 1 cm3 .

in Fig. 1. The system consists of a signal generator (SG; 33210A, Agilent), a 50-dB gain power amplifier (BSA0110100, BONN Elektronik), a power sensor at the receiving port, a 10-dB attenuator at the transmitting side to protect the signal port of the SG from an excessive power reflected from the WPT system, a 30-dB attenuator at the receiving side to attenuate the power flowing into the power sensor, a PC for acquiring data, a 3-D axis positioner, and WPT systems. The magnetic near-field probes shown in Fig. 2 were developed in-house and used to measure simultaneously both the magnitude and phase of magnetic fields. The probes consist of three rectangular loops oriented orthogonally for the measurement of the three x-, y-, and z-components of the magnetic field. The signals in the time domain received by each rectangular loop are amplified by a preamplifier and converted into digital data in a time sequence by an analog/digital converter. The magnitude and phase of the magnetic field are finally determined by the fast Fourier transform of the acquired time-sequence data. The probes are calibrated before the near-field measurement, and the correction factor for finding the amplitude of the magnetic field is calculated by the three-antenna method using a loop antenna [17]. The calibration factor of the magnetic-field probe was 10.97 μT/mV at 100 kHz. The linearity is less than 0.5 dB for the frequency range between 20 kHz and 10 MHz. The isotropic deviation is less than 0.3 dB at 100 kHz, and the dynamic range is from 0.04 to 240 A/m. A magnetic-field probe is scanned to measure the magnetic field inside the measurement area while the other magnetic-field probe is fixed at a specific location to obtain the reference phase for each measurement. The three-axis positioner is connected to the PC and controlled via a general-purpose interface bus cable. B. WPT Systems Fig. 3(a) and (b) illustrates a schematic view of WPT systems without and with ferrite tiles, respectively. The dimensions of the transmitting and receiving coils are set as follows: the inner and outer radii are a = 120 mm and b = 225 mm,

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Fig. 4. Power transmission ratio of fabricated WPT systems without (black) and with (red) ferrite tiles.

Fig. 3. Schematic view of WPT systems (a) without and (b) with ferrite tiles. (c) Fabricated WPT system without ferrite tiles used in the measurement.

respectively, and each coil is wound in N = 44 turns. The transmitting and receiving coils are both loaded with a capacitance of C f = C L = 2.1 nF in order to tune the resonant frequencies of the system. A terminal load of R L = 50  is connected at the end port of the receiving coil. The spacing between the transmitting and receiving coils is fixed to d = 200 mm in all measurements. Fig. 3(c) shows our fabricated WPT system without ferrite tiles (TDK, IB-017). For the system with the ferrite tiles, metal plates made of aluminum with a size of 600 mm × 600 mm are placed at the back of the ferrite tiles, as shown in Fig. 3(b). The thicknesses of the ferrite tiles and metal plates are 5 and 2 mm, respectively. The spacing between coils and the ferrite tiles is 10 mm. The resonance frequencies of our fabricated WPT system without ferrite tiles are measured by using a vector network analyzer (VNA; E5071C, Agilent) as 111 and 125 kHz. Therefore, the power transmission ratio is calculated as η21 ≡

PL = |S21 |2 × 100[%] Pinc

(1)

where PL and Pinc denote the power dissipated in the load and the incident power applied to the system, respectively. Note that the power transfer efficiency of the system is calculated using the ratio of the output power to the input power [18]. In this paper, we used the incident power as the denominator in (1) instead. Fig. 4 shows the measured power transmission ratio for the WPT system without ferrite tiles, which was approximately 96% at both frequencies. Although not shown here, the numerical results of the efficiency are in good agreement with those measured by the VNA with a difference of less than 2%. The power transmission ratios of the WPT system with the ferrite tiles were measured as 79% and 85% at 80 and 97 kHz, respectively. When the WPT system without

the ferrite tiles [Fig. 3(a)] is placed close to the metal ground plane, the power transfer efficiency decreases to less than 20%, which is almost impractical for the use of WPT, whereas the power transfer efficiency is kept as high as 80% by using the ferrite tiles to shield the leakage magnetic flux. It was indicated that the resonant frequency shifted to the lower one when the ferrite tiles were attached owing to an increase in inductance in the WPT system. The currents flowing in the transmitting and receiving coils of the system with no ferrite tiles were determined numerically as 139 and 138.9 A(rms) for an input power of 1 W at 125 kHz, while those measured by a power analyzer (PA3000, Tektronix) were 138.2 and 138.9 A(rms), respectively. The difference in the simulated and measured currents was less than 1%. For the WPT system with ferrite tiles, the currents in the transmitting and receiving coils were measured as 112.8 and 128.4 A(rms), respectively, for 1-W input power at 97 kHz. In a real WPT system, the load resistance is not always 50  as used in this paper. The load may also vary depending on the state of charging. However, in such a case, we can still apply our proposed method and measurement system by monitoring the currents flowing in the transmitting and receiving coils, and recording their values at each measurement for normalization, since strength of the radiated EM fields depends on the currents. III. H UMAN -B ODY M ODELS AND E XPOSURE S ITUATIONS A. Numerical Human-Body Models Realistic human models of the Japanese adult male “TARO” developed by the National Institute of Information and Communications Technology, Tokyo, Japan, were used in numerical simulations [19]. We apply this model since it represents an average 18- to 30-year-old Japanese male. The adult male model was developed from high-resolution MRI data and comprises 51 different tissues and organs with a resolution of 2 mm. The height and weight of the model are 173 cm and 64 kg, respectively. The permittivity and conductivity of the tissues were extracted from Gabriel’s database [20]. The dielectric constants for the 51 tissues are also indicated in the Appendix. The analysis frequency is set to 125 kHz, which

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magnetic field induces an electromotive force in accordance with Faraday’s law as   d (5) Hz |i, j,k d yd x Vz |i, j,k = −μ0 dt = − j ωμ0 Hz |i, j,k l x l y (6)

Fig. 5. Impedance network that represent a human body. Z x |i, j,k , Z y |i, j,k , and Z z |i, j,k are the impedances of the i, j, kth voxels in the x-, y-, and z-directions, respectively.

is the resonant frequency of the WPT system without ferrite tiles while it is set to 97 kHz for the system with ferrite tiles. In this paper, a new approach combining experiments and a numerical method is used. First, magnetic near-field data are measured by the near-field measurement system using the magnetic-field probes. Then, the measured magnetic fields are used as incident fields in the impedance method to derive the internal electric field strength inside the human-body model. The impedance method was first developed for calculation in 2-D space by Gandhi et al. [21] then extended to 3-D space [22]. Since the MQS approximation is valid in this frequency region, the impedance method is also applicable to this problem [10], [23]. B. Impedance Method In the impedance method, an inhomogeneous human-body model is modeled as a 3-D impedance network, as shown in Fig. 5. Each voxel is associated with dielectric constant corresponding to the location in the human-body model. An impedance is assigned at each edge of the voxel. Therefore, the impedance at each edge is represented by an average of the dielectric constants of four adjacent voxels, i.e., for an impedance along the x-direction Z x Z x |i, j,k =

1 lx j ωε0 ε˙ a |i, j,k l y l z

(2)

where ω and ε0 are the angular frequency and permittivity in free space, respectively. l x , l y , and l z are the edge lengths in the x-, y-, and z-directions, respectively. ε˙ a denotes the average complex relative permittivity, that is, ε˙ |i, j,k + ε˙ |i, j +1,k + ε˙ |i, j,k+1 + ε˙ |i, j +1,k+1 4 σ |i, j,k = εr |i, j,k + j ωε0

ε˙ a |i, j,k = ε˙ |i, j,k

(3) (4)

where εr |i, j,k and σ |i, j,k are the relative permittivity and conductivity, respectively, associated with each voxel at the location indexes i , j , and k. Impedances aligned in the y- and z-directions can be derived in a similar way. Assuming a loop current along the edge of the voxel in the x y plane; incident

where μ0 is the permeability in free space, and Hz denotes the incident magnetic field in the z-direction, which is measured using the proposed system. Note that the location of Hz is at the center of the voxel face. Since each loop current passes through four impedances and the line current through the impedance at each edge is the sum of four loop currents, in the x y plane, we obtain the following equation: Z x |i, j,k [Iz |i, j,k − Iz |i, j −1,k − I y |i, j,k + I y |i, j,k−1 ] + Z y |i, j,k [Iz |i, j,k − Iz |i−1, j,k − Ix |i, j,k + Ix |i, j,k−1 ] + Z x |i, j +1,k [Iz |i, j,k − Iz |i, j +1,k + I y |i, j +1,k − I y |i, j +1,k−1 ] + Z y |i+1, j,k [Iz |i, j,k − Iz |i+1, j,k + Ix |i+1, j,k − Ix |i+1, j,k−1 ] = Vz |i, j,k

(7)

where Ix |i, j,k , I y |i, j,k , and Iz |i, j,k are the loop current in the yz, x z, and x y planes, respectively. We can construct similar equations for the electromotive force in the x y and yz planes. After solving the system of equations by the successive overrelaxation method, we can obtain the solutions of the loop currents, and then we obtain the line currents along the edges of each voxel by adding the values of the four loop currents surrounding each edge. We can estimate the current at the center of each voxel by averaging the four line currents in each direction. Internal electric field is then determined as, for example, for the z-component E zin |i, j,k =

σ |i, j,k

Izc |i, j,k 1 + j ωε0 εr |i, j,k l x l y

(8)

where Izc |i, j,k is the current along the z-direction at the center of the voxel at the location indexes i , j , and k. Our impedance method has been validated by comparing the results obtained for a dielectric sphere with Mie’s analytical solution [24]. A comparison of the internal electric field in the Japanese male model induced by the exposure of a low-frequency magnetic field was also shown previously [25], and it was found that the relative difference in the maximum internal electric field was less than 30% for our numerical code. C. Exposure Conditions Exposure was numerically simulated by using the measured magnetic-field distribution as an incident magnetic field in the impedance method. Although the 99th percentile value is suggested for use in an international guideline [7], the 99.9th percentile electric field (E99.9) is used as a metric from the viewpoint of conservativeness since the exposure is highly nonuniform [26]. In addition, the 99th percentile electric field (E99 ) is also determined for each exposure condition. Fig. 6 illustrates the exposure condition assuming that the WPT systems are used for charging EVs and placed on a metal ground plane. The distance from the WPT systems to the measurement volume was varied from 200 to 300 mm

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Fig. 6. Exposure conditions for WPT coils which are placed (a) under a free-space condition without ferrites and (b) on a ground with ferrites and a metal plate as a vehicle body. Note that all measurements were carried out without a human body inside the volume.

Fig. 8. (a) Amplitude and (b) phase distributions of magnetic field measured on the yz plane at 200-mm distance from the WPT system without the ferrite tiles and those obtained by MoM for an input power of 1 W.

IV. R ESULTS AND N UMERICAL A SSESSMENT Fig. 7. Measurement of magnetic-field distribution close to WPT system on the ground with ferrites and metal plate for the exposure condition shown in Fig. 6(b).

for a free-space condition in Fig. 6(a), while the distance was varied from 235 to 400 mm due to the size of the metal plate simulating bottom of the vehicle body, as shown in Fig. 6(b). The metal plate is made of aluminum. The size of the metal plate is 1200 mm × 1200 mm. The distance from the outer edge of the coil to the edge of the metal plate is 200 mm. The input power of the system is 1 W. Fig. 7 shows the measurement of the magnetic-field distribution for the WPT system with ferrite tiles and a metal plate. The size of the measurement volume is 350 mm × 700 mm × 1800 mm to include the entire human-body model. The measurement resolution is 50 mm; thus, the total number of measurement points is 8 × 15 × 37 = 4440. The measurement time required for a measurement point (including the time needed to move the magnetic-field probe to the next measurement point and the waiting time of 2 s after the probe stops at the next measurement point) was about 9 s. Since the measurement was carried out from the z-axis (vertical line), it is also necessary to move the probe from the topmost measurement point to the bottommost position, which requires about 40 s. The total measurement time required for one measurement is 9 × 4440 + 40 × 8 × 15 = 44 760 s or 12 h 26 min. In addition, note that all measurements were performed without a human body.

To begin with, the validity of the proposed approach of combining measurement data of the magnetic near-field with the impedance method is confirmed by comparison with numerical results. A. Validation of Numerical WPT Model Fig. 8(a) and (b) shows the distributions of the amplitude and phase of the magnetic field measured and numerically obtained by the method of moments (MoMs), respectively, on the yz plane at 200-mm distance from the WPT coils for the free-space condition [see Fig. 6(a)]. It is obvious that both numerical and experimental results are in good agreement with each other. The error in the phase was larger when the magnitude of the measured magnetic fields was smaller owing to the deterioration of the signal-to-noise ratio of the system. The maximum difference in the magnitudes obtained by MoM and measurements was approximately 7.9%, proving the validity of our near-field measurement system. The difference in magnitude may be due to the fabrication error of the WPT system and the measurement uncertainty of the magnetic-field probes. The uncertainty for the probes was determined using the square root of the sum of the squares of the linearity and isotropic errors of the probe to be approximately 7% (0.58 dB), which is close to the maximum error of 7.9% in the measurement. The other cause of the error may be due to the positioning of the probe. It was also found that there are two or more peaks in the distribution

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Fig. 9. Distribution of the induced electric field strength for adult male model for an input power of 7.7 kW. The distance from the models to the WPT system is 200 mm.

of the measured magnetic near field, which were aligned in opposite directions, i.e., with a 180° phase difference. Each peak of the magnetic field may contribute to the internal electric field in an additive or subtractive manner. Therefore, for accurate exposure evaluation, it is necessary to obtain the phase information of the magnetic near field. B. Dosimetry of WPT System Without Ferrite Tiles Next, measured or calculated magnetic fields are used as incident fields in the impedance method. As mentioned in the previous section, this two-step method is only applicable when the MQS approximation is valid. It has been shown [24] that the error of the internal electric field (maximum values for 2-mm resolution) when neglecting the incident electric field for a plane-wave exposure to a dielectric sphere is approximately 39% at 100 kHz, and the error decreases as the E/H ratio of the incident field becomes smaller. For our experiment, the incident electric fields measured with an electric field probe (FL7030, Amplifier Research Co.) were 45.7 and 41.8 V/m for the WPT systems without and with ferrite tiles, respectively, at 1-W input power, corresponding to E/H ratios of 70 and 131 . The errors in the internal electric field when neglecting the incident electric field were estimated to be less than 30% for the maximum value and 18% for the average value at 100 kHz [24]. Note also that the resolution of the human-body models is 2 mm while the measurement interval is 50 mm. Therefore, a linear interpolation of the measurement data into 2-mm resolution must be carried out before using them in numerical analyses. Computational times required for finding internal electric field distribution by using the impedance method were approximately 13 h. Fig. 9 shows the distribution of the internal electric field strength when the input power is P in = 7.7 kW under a free-space condition [see Fig. 6(a)]. The distribution when the human body was exposed from the front is also shown. The distance from the outermost extension of the WPT coils to the nearest part of the human body (the heel for exposure from the back and the toe for exposure from the front) is set as L = 200 mm. From Fig. 9, it is seen that the internal electric field strength was found to be highest at the ankle of the model when the human body was exposed from the back,

Fig. 10. Distributions of (a) magnitude and (b) phase of the magnetic-field strength for WPT coils without (left) and with (right) the ferrite tiles. The input power is normalized to 7.7 kW in the xz plane.

while the highest value is at the crotch when the human body was exposed from the front. E99.9 and E99 were derived as 2.13 and 1.40 V/m, respectively, when the human body was exposed from the back at P in = 7.7 kW. E99.9 and E99 were 1.61 and 0.80 V/m, respectively, for the exposure from the front. Therefore, the exposure from the back results in higher exposure levels, giving a conservative evaluation. These values were lower than the basic restriction of the internal electric field strength at 125 kHz, which is equal to 16.875 V/m in the ICNIRP guideline [7], whereas the magnetic-field strengths of 57 A/m was higher than the reference level (21 A/m). The results imply that the compliance with the basic restriction is mandated in order to apply an input power of 7.7 kW for this WPT system. The differences in E99.9 and E99 calculated by using the MoM-derived magnetic field and those using the measurement data were less than 20%. C. Dosimetry of WPT System With Ferrite Tiles and Metal Vehicle Body For the compliance of an actual WPT system, information on the detailed shape, type, or dimensions of WPT coils may

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Fig. 12. Coupling factor derived by using measured magnetic field of WPT coils without the ferrites and with the ferrites. Solid line shows numerical results using simulated magnetic field for WPT coils without the ferrites.

Fig. 11. Induced electric field strength of the adult male model and the current flowthrough a cross-sectional area for the WPT coils without and with the ferrites. Distance between the model and WPT coils is L = 235 mm.

not be provided by manufacturers. In some case, it is also difficult to accurately model an actual exposure condition, including a vehicle body, vehicle tires, and so on. Therefore, it is difficult to perform dosimetry using only a numerical approach in such cases. In this paper, the combination of measured data and the numerical approach of the impedance method are used. First, magnetic-field strengths, as well as their phases, are measured. Then, they are used as an incident field in the impedance method in the second step as in Section III-B. Fig. 10(a) and (b) shows distributions of the measured magnetic-field strength in the x z plane under a free-space condition and for the WPT system equipped with ferrite tiles, corresponding to the exposure scenarios in Fig. 6(a) and (b), respectively. The results are indicated for an input power of P in = 7.7 kW. Note that a metal plate with a size of 1200 mm × 1200 mm is placed above the ferrite tiles of the WPT system to consider the shielding effect of a vehicle body. From the results, it is obvious that the magnetic-field strength decreases to almost zero at a measurement position higher than the metal plate due to the shielding effect. It was also found that the magnetic-field strength of the WPT system equipped with the ferrites was smaller than that without the ferrite tiles and that its maximum was 28 A/m, when L = 235 mm. Even for the WPT system shielded with ferrite tiles, the magnetic-field strength exceeds the reference level. Therefore, the compliance with the basic restriction is required. Fig. 11 shows the distribution of the internal electric field strength and the z-directed current flowing through a crosssectional area of the human body for WPT coils without and with ferrite tiles, when L = 235 mm. The current for the kth cross-sectional layer of the human model can be calculated by summing the currents in all the voxels in that layer by using the following equation:  σ |i, j,k · E zin |i, j,k . (9) Ilayer (k) = l x l y i, j

The results were calculated by using the measured magnetic fields. As can be seen from Fig. 11, the internal field concentrates around the legs since only the bottom part of the models was exposed to the magnetic fields. E99.9 and E99 were 1.92 and 1.21 V/m, respectively, for the WPT systems without ferrite tiles, whereas they were 0.42 and 0.26 V/m, respectively, for the WPT system with the ferrite tiles. It is also observed that the maximum current flows at the crotch of the body for the WPT coils without the ferrite tiles, whereas the maximum appears at the mid-calf of the human leg since the magnetic-field distribution was highly nonuniform for the WPT coils with the ferrites. D. Discussion on Coupling Factor and Compliance of WPT Systems The coupling factor is a concept adapted from the IEC 62233 and 62311 standards for the evaluation of product safety due to human exposure [27], [28]. It represents the relationship between the incident magnetic field and the internal electric field strength inside a human body. For the WPT systems in the intermediate frequency band, the MQS approximation was applicable and the coupling factor is also valid in this frequency band [29]. This concept was specifically developed for low-frequency nonuniform exposure and extended to also consider the thermal effect limit by Tetsu et al. [30]. Wake et al. [31] also derived the coupling factors for various WPT systems. The coupling factors normalized with the reference level of the magnetic field and basic restrictions of the internal electric field strength are defined by the following equation [30], [31]:     in i E 99.9 Hmax / (10) αc,E = in i E limit Hlimit in i where E 99.9 and Hmax denote the 99.9th percentile value of the internal electric field strength (averaging volume of 2 mm × 2 mm × 2 mm) and the maximum spatial value of the incident magnetic-field strength in a volume occupied by a human body model, respectively. In this paper, an averaging area of 100 cm2 is applied for the calculation of the magneticfield strength in accordance with the IEC standard [27].

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TABLE I D IELECTRIC C ONSTANTS OF B IOLOGICAL T ISSUES AT 125 kHz FOR A DULT M ALE M ODEL U SED IN T HIS PAPER

in i E limit and Hlimit are the basic restrictions of the internal electric field strength and the reference level of the magnetic-field strength, respectively, which are prescribed in the guidelines. The coupling factors defined in (1) can be derived from numerical simulations or experiments. When measuring the i magnetic-field strength for a practical compliance test, Hmax in (1) can be simply changed to the measured value. It is also obvious that the denominator in (1) has a constant value and the coupling factor αc,E has no unit. The advantages of using the coupling factor are that once a conservative value of this factor has been determined, the compliance test can be carried out by multiplying the spatial peak value of the magnetic-field strength by the coupling factor, and the derived values are compared directly with the reference levels. However, the drawback of the coupling factor is that its value is difficult to generalize because the size, shape, materials used, and exposure conditions for each WPT system are different. For EVs, the location of WPT systems and the exposure conditions are actually well defined. In this case, we may define a coupling factor for use in the compliance of the WPT systems for EVs. Fig. 12 shows the coupling factors determined by using the measured magnetic-field distribution for the adult human model at different distances. Also, Fig. 12 shows the coupling factors determined by using the MoM-derived magnetic-field distribution. It was demonstrated that both numerical and experimental results are in good agreement, having a difference of less than 5%. The coupling factors of the WPT coils without the ferrite are also higher than those of the WPT coils with the ferrite since the field distribution is more distributed for the case without the ferrites. In addition, it was observed that the coupling factor tends to increase when the distance from the WPT coils increases for all models. This is because the field distribution becomes more uniform when the distance increases. The maximum value of the coupling factor is 0.068, when L = 300 mm. This value was derived for the WPT system placed in free space [see Fig. 6(a)]. For the realistic exposure scenario of the WPT system with the ferrite tiles and the metal plate on the ground

plane, the maximum coupling factor was approximately 0.037, when L = 400 mm. Note also that these values are much lower than 0.33, which is the value derived by using a uniform incident field. The compliance with the reference level of the magnetic-field strength yields the maximum allowable input power (MAIP) as 1585 and 4343 W for the WPT coils without and with the ferrites, respectively, when L = 235 mm. Hence, the compliance with the reference level is not satisfied for P in = 7.7 kW. The compliance using the coupling factor can be carried out by multiplying the coupling factor by the measured magnetic-field strength and then comparing the result with the reference level. For our cases, the coupling factor at L = 235 mm was determined as 0.051. Therefore, it allows an MAIP of approximately (1/0.051)2 = 385 times or approximately 26 dB, compared with those determined by the reference level. The coupling factor for the WPT system with the ferrite tiles is 0.024, which is 2.1 times smaller than that of the WPT system without the ferrite tiles at L = 235 mm. In some cases, a WPT system is equipped with components for EM countermeasures, such as ferrites, making it difficult to model accurately in numerical simulations. By using the proposed method, we are still able to conduct accurate exposure evaluations of actual WPT systems with such WPT sources or when no information of the coil structures or details of the dimensions are provided. V. C ONCLUSION In this paper, a combined approach of measurement and numerical analysis using the impedance method was proposed to compute the internal electric field strength in an adult human model for exposure assessment. The magnetic-field strength emitted from the WPT systems was measured using our near-field measurement system with in-house-developed magnetic-field probes. It was found that both numerical and experimental results of the magnetic-field distributions were in good agreement with each other. Then, the measured field distributions were used to derive the internal electric field

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strength and the coupling factor in the human body models. The coupling factor was smaller for the WPT system with the ferrites since the magnetic-field distribution was more compact or more nonuniform owing to the shielding effect of the ferrites. In conclusion, the proposed approach is useful and practical for the exposure assessment and compliance of WPT systems operating in a low-frequency band, especially for the case that no information of the coil structures or details of the dimensions are provided. A future subject is to develop a probe-array system to reduce the measurement time. From the viewpoint of compliance, the consideration of exposure for various numerical human models is also required. Moreover, the variability when applying different human model poses and the use of different systems such as double D coils, should be determined to generalize the coupling factor for the WPT system of EVs for use in practical compliance. A PPENDIX The dielectric constants at 125 kHz used in the numerical analyses in this paper are given in Table I. The values at 97 kHz are substantially similar to those at 125 kHz; thus, they are not indicated here. ACKNOWLEDGMENT Part of the numerical results was obtained using the SX-ACE Supercomputer at Tohoku University, Sendai, Japan. The authors would also like to thank a previous graduate student, Y. Aoki, from the Tokyo University of Agriculture and Technology, Tokyo, Japan, for conducting numerical dosimetry and performing measurements. R EFERENCES [1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljaˇci´c, “Wireless power transfer via strongly coupled magnetic resonances,” Science, vol. 317, no. 5834, pp. 83–86, 2007. [2] N. Shinohara, “Power without wires,” IEEE Microw. Mag., vol. 12, no. 7, pp. S64–S73, Dec. 2011. [3] L. Showa. Wireless power supply. Aircraft Industry Co. Accessed: Sep. 9, 2017. [Online]. Available: http://www.showa-aircraft.co.jp/ business/products/kyuuden/ [4] S. Li and C. C. Mi, “Wireless power transfer for electric vehicle applications,” IEEE J. Emerg. Sel. Topics Power Electron., vol. 3, no. 1, pp. 4–17, Mar. 2015. [5] S. Choi, J. Huh, W. Y. Lee, S. W. Lee, and C. T. Rim, “New cross-segmented power supply rails for roadway-powered electric vehicles,” IEEE Trans. Power Electron., vol. 28, no. 12, pp. 5832–5841, Dec. 2013. [6] A. Ahlbom et al., “Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 GHz),” Health Phys., vol. 74, no. 4, pp. 494–521, Apr. 1998. [7] International Commission on Non-Ionizing Radiation Protection, “Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz),” Health Phys., vol. 99, pp. 818–836, Dec. 2010. [8] IEEE Standard for Safety Levels With Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEE Standard C95.1-1999, 1999. [9] IEEE Standard for Safety Levels With Respect to Human Exposure to Electromagnetic Fields, 0–3 kHz, IEEE Standard C95.6-2002, 2002. [10] I. Laakso, S. Tsuchida, A. Hirata, and Y. Kamimura, “Evaluation of SAR in a human body model due to wireless power transmission in the 10 MHz band,” Phys. Med. Biol., vol. 57, pp. 4991–5002, Jul. 2012.

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[11] S. W. Park, K. Wake, and S. Watanabe, “Incident electric field effect and numerical dosimetry for a wireless power transfer system using magnetically coupled resonances,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3461–3469, Sep. 2013. [12] T. Sunohara, A. Hirata, I. Laakso, and T. Onishi, “Analysis of in situ electric field and specific absorption rate in human models for wireless power transfer system with induction coupling,” Phys. Med. Biol., vol. 59, pp. 3721–3735, Jun. 2014. [13] T. D. Bracken, “Assessing compliance with power-frequency magnetic-field guidelines,” Health Phys., vol. 83, pp. 409–416, Sep. 2002. [14] M. A. Stuchly and T. W. Dawson, “Human body exposure to power lines: Relation of induced quantities to external magnetic fields,” Health Phys., vol. 83, pp. 333–340, Sep. 2002. [15] J. Swanson, “A transmission utility’s experience of applying EMF exposure standards,” Health Phys., vol. 83, no. 3, pp. 417–425, 2002. [16] K. Wake et al., “Magnetic fields in intermediate frequency band generated by IH-hobs,” in Proc. 34th Annu. Meeting Bioelectromagn. Soc., vol. 91, Brisbane, QLD, Australia, 2012, p. 118. [17] M. Ishii and K. Fujii, “Loop antennna calibration methods in lowfrequency,” in Proc. Int. Symp. Electromagn. Compat. (EMC/Tokyo), Tokyo, Japan, May 2014, pp. 290–293. [18] Q. Yuan, Q. Chen, L. Li, and K. Sawaya, “Numerical analysis on transmission efficiency of evanescent resonant coupling wireless power transfer system,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1751–1758, May 2010. [19] T. Nagaoka et al., “Development of realistic high-resolution wholebody voxel models of Japanese adult males and females of average height and weight, and application of models to radio-frequency electromagnetic-field dosimetry,” Phys. Med. Biol., vol. 49, pp. 1–15, Dec. 2004. [20] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, no. 11, p. 2271, 1996. [21] O. P. Gandhi, J. F. Deford, and H. Kanai, “Impedence method for calculation of power deposition patterns in magnetically induced hyperthermia,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 10, pp. 644–651, Oct. 1984. [22] N. Orcutt and O. P. Gandhi, “A 3-D impedance method to calculate power deposition in biological bodies subjected to time varying magnetic fields,” IEEE Trans. Biomed. Eng., vol. BME-35, no. 8, pp. 577–583, Aug. 1988. [23] I. Laakso, T. Shimamoto, A. Hirata, and M. Feliziani, “Quasistatic approximation for exposure assessment of wireless power transfer,” IEICE Trans. Commun., vol. E98-B, no. 7, pp. 1156–1163, Jul. 2015. [24] S. W. Park, K. Wake, and S. Watanabe, “Calculation errors of the electric field induced in a human body under quasi-static approximation conditions,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 5, pp. 2153–2160, May 2013. [25] A. Hirata et al., “Intercomparison of induced fields in Japanese male model for ELF magnetic field exposures: Effect of different computational methods and codes,” Radiat. Protection Dosimetry, vol. 138, pp. 237–244, Nov. 2010. [26] L. Ilkka and H. Akimasa, “Evaluation of the induced electric field and compliance procedure for a wireless power transfer system in an electrical vehicle,” Phys. Med. Biol., vol. 58, no. 21, p. 7583, 2013. [27] Measurement Methods for Electromagnetic Fields of Household Appliances and Similar Apparatus With Regard to Human Exposure, Standard IS/IEC 62233, IEC, Geneva, Switzerland, 2005. [28] Assessment of Electronic and Electrical Equipment Related to Human Exposure Restrictions for Electromagnetic Fields (0 Hz–300 GHz), Standard IEC62311, IEC, Geneva, Switzerland, 2007. [29] A. Hirata, I. Fumihiro, and I. Laakso, “Confirmation of quasi-static approximation in SAR evaluation for a wireless power transfer system,” Phys. Med. Biol., vol. 58, no. 17, p. N241, 2013. [30] S. Tetsu, H. Akimasa, L. Ilkka, V. De Santis, and O. Teruo, “Evaluation of nonuniform field exposures with coupling factors,” Phys. Med. Biol., vol. 60, no. 20, p. 8129, 2015. [31] K. Wake et al., “Derivation of coupling factors for different wireless power transfer systems: Inter- and intralaboratory comparison,” IEEE Trans. Electromagn. Compat., vol. 59, no. 2, pp. 677–685, Apr. 2017.

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Jerdvisanop Chakarothai (S’06–M’10) received the B.E. degree in electrical and electronic engineering from Akita University, Akita, Japan, in 2003, and the M.E. and D.E. degrees in electrical and communication engineering from Tohoku University, Sendai, Japan, in 2005 and 2010, respectively. He joined the Nagoya Institute of Technology, Nagoya, Japan, in 2011, and Tokyo Metropolitan University, Tokyo, Japan, in 2013. He has been a Research Associate with Tohoku University since 2010. He is currently with the National Institute of Information and Communications Technology, Tokyo. His current research interests include computational electromagnetics (CEM) for biomedical communications and electromagnetic compatibility. Dr. Chakarothai is a member of the Institute of Electronics, Information and Communication Engineers, Japan, and the Institute of Electrical Engineers, Japan. He is also a member of the Bioelectromagnetic Society and the Applied CEM Society. He was the recipient of the 2014 Young Scientist Award from the International Scientific Radio Union. Kanako Wake (M’05) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1995, 1997, and 2000, respectively. She is currently with the National Institute of Information and Communications Technology, Tokyo, where she is involved in research on biomedical electromagnetic compatibility. Dr. Wake is a member of the Institute of Electronics, Information and Communication Engineers, the Institute of Electrical Engineers, Japan, and the Bioelectromagnetics Society. She was a recipient of the 1999 International Scientific Radio Union Young Scientist Award. Takuji Arima (M’04) received the M.E. and D.E. degrees in engineering from the Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan, in 1999 and 2002, respectively. He is currently an Associate Professor with the Department of Electrical and Electronics Engineering, TUAT, and also a Part-Time Researcher with the National Institute of Information and Communications Technology, Tokyo. His current research interests include computational electromagnetics and metamaterials. Dr. Arima was the recipient of the Young Scientist Award from the IEEE Antennas and Propagation Society Japan Chapter.

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Soichi Watanabe (S’93–M’96) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1991, 1993, and 1996, respectively. He is currently with the National Institute of Information and Communications Technology, Tokyo. His current research interest includes biomedical electromagnetic compatibility. Dr. Watanabe is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers, Japan, and the Bioelectromagnetics Society. From 2004 to 2012, he was a member of the Standing Committee III on Physics and Engineering, International Commission on Non-Ionizing Radiation Protection (ICNIRP). He he has been a member of the main commission of ICNIRP since 2012. He was the recipient of the 1996 Young Scientist Award from the International Scientific Radio Union, the 1997 Best Paper Award from the IEICE, and the 2004 Best Paper Award (The Roberts Prize) of Physics in Medicine and Biology.

Toru Uno (M’85–SM’02) received the B.S.E.E degree from the Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan, in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1982 and 1985, respectively. He was a Research Associate with the Department of Electrical Engineering, Tohoku University, from 1985 to 1991, and an Associate Professor from 1991 to 1994. From 1998 to 1999, he was on leave from the TUAT. He was a Visiting Scholar with the Department of Electrical Engineering, Pennsylvania State University, State College, PA, USA. He is currently a Professor with the Department of Electrical and Electronics Engineering, TUAT. Dr. Uno served as the Chair of the IEEE Antennas and Propagation Society Japan Chapter from 2005 to 2006 and an Associate Editor of the IEEE A NTENNAS AND W IRELESS P ROPAGATION L ETTERS from 2008 to 2013. He was the Chair of the Technical Group on Antennas and Propagation, Institute of Electronics, Information and Communications Engineers (IEICE) from 2011 to 2012, and was a General Chair of the 2016 International Symposium on Antennas and Propagation. He is currently a Fellow of the IEICE. He was the recipient of the Young Scientist Award, the Distinguished Contributions Award, and the Paper Award from the IEICE in 1990, 2006, and 2007, respectively.

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A Planar Dipole Array Surface for Electromagnetic Energy Harvesting and Wireless Power Transfer Ahmed Z. Ashoor, Member, IEEE, Thamer S. Almoneef, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract— We present a design of an electromagnetic energy harvesting surface inspired by an array of printed metallic dipolar elements. The unit cell of the proposed harvester is based on two printed asymmetric off-center fed dipoles. As a proof of concept, a finite array of 9×3 unit cells was analyzed numerically and experimentally. The array was first analyzed for maximizing radiation to ac absorption where each dipole was terminated by a resistor across its gap. For dc conversion, the resistors were the replaced by Schottky diodes. The simulation results show radiation to ac harvesting efficiency of 99%. An overall radiation to dc harvesting efficiency of 76% was obtained experimentally, which, to the best of the authors’ knowledge, exceeds the performance of all previous energy harvesting surfaces. Another critical feature of the proposed designs is enhancing the power per diode in order to maximize its turn-ON time. Index Terms— Dipole array, electromagnetic absorber, energy harvesting, rectennas, rectification, wireless power transfer (WPT).

I. I NTRODUCTION

I

N THE early years of the 20th century, the concept of transferring power wirelessly using electromagnetic waves was first experimented by Nikola Tesla [1]. In the 1960s, Brown [1] demonstrated the concept of the far-field wireless power transfer (WPT) using radio frequency (RF) electromagnetic waves as a source of energy. Brown [1] designed a rectifying system, the rectenna, to convert microwaves to dc power and was the first to successfully demonstrate a rectenna system for wirelessly powering an aircraft. Later, Glaser [2] introduced the idea of using the concept of WPT to harvest energy from space. Space solar power is based on using solar cells in space to collect solar energy and then converting the power to microwaves for transmission to earth via highly directive antennas. Microwave energy is then received on the earth surface utilizing large rectenna farms that receive and convert the microwave energy to usable power. Electromagnetic energy harvesting and WPT refer to capturing the energy/power of electromagnetic waves arriving from a distant source using rectennas (i.e., antenna with rectifying circuitry). Many single rectennas operating in the microwave bands were developed having a radiation to dc conversion Manuscript received March 13, 2017; revised July 19, 2017; accepted August 25, 2017. (Corresponding author: Omar M. Ramahi.) A. Z. Ashoor and O. M. Ramahi are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L3G1, Canada (e-mail: [email protected]; [email protected]). T. S. Almoneef is with the Department of Electrical Engineering, Prince Sattam University, Al Kharj 16278, Saudi Arabia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2750163

efficiency reaching 80% [3]. However, the conversion efficiency for rectennas when incorporated in an array is not as high as the single rectenna case. Several researchers have designed energy harvesting arrays using primarily metallic antennas such as microstrip patches, spiral antennas, and metamaterial cells [3]–[11]. Most of these energy harvesting arrays provide either the radiation to ac efficiency or the RF to dc conversion efficiency. In 2001, planner arrays using spiral antenna were designed for harvesting ambient power with a maximum radiation to dc efficiency of 45% and 35%, respectively [4]. In recent works, metasurfaces were introduced for energy harvesting with the primary objective of maximizing the harvested energy per footprint [9]–[11]. Although the metasurface type of harvester provided high radiation to ac/dc conversion efficiency, the level of power density available at each element was low since the elements are inherently electrically small [9], [10], [12]–[14]. While maximizing the radiation to ac/dc conversion efficiency is a very important goal in designing energy harvesters as demonstrated by recent works [15], [16], an equally important consideration is maximizing the energy at the feed of all rectification diodes that are used in the harvester. Considering that a voltage threshold is required for any diode to turn ON, maximizing the power available at each of the diodes of the harvester maximizes the turn-ON time, which in turn maximizes the overall power transfer to the load. Despite the fact that different single rectenna structures have been studied in the literature using different types of antenna and rectification circuitry [17]–[20], single rectennas are only capable of capturing very low electromagnetic energy. However, more electromagnetic energy can be harvested using multiple antennas in an array form [21]. Challenges are raised as how to combine the harvested/received power from multiple array antennas/collectors in a rectenna array. Combining RF and dc power has been studied experimentally using different rectenna arrays [22]–[27]. The RF combining structure uses RF circuits to deliver all incoming RF power into a single rectifying circuit thereby increasing the overall diode efficiency. This approach is complex and may require additional layer for the RF combiner circuitry while incurring RF transmission losses. On the other hand, dc combining in which each antenna/collector has a rectifying circuit is realized by adding the rectified dc power using either a voltage or current summation or combinations of both. This approach does not require a complex RF network and it has architectural simplicity and modularity. In this paper, our goal is twofold. First, design an energy harvesting planar surface based on printed circuit board

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technology that has high microwaves to ac/dc conversion efficiency. While most of the pervious works that show high energy harvesting efficiencies were focused on single rectennas [3], we introduce an array of rectennas that is designed to have a near unity radiation to ac conversion efficiency. Second, the design ensures that the harvester provides high power density per diode to maximize diode turn-ON time over the duration of operation. To maximize the energy density per diode, we started with harvesting elements that are not electrically small. For channeling the dc current, we introduced a channeling mechanism that places the rectification diode on the same layer of the printed dipoles. This avoids the need for an RF or dc combining circuit that requires additional layers. Finally, the proposed harvester shows less sensitivity to incident angles. II. D ESIGN M ETHODOLOGY OF THE H ARVESTER U NIT C ELL In light of our goal of increasing the power density per diode while achieving high microwaves to ac/dc conversion efficiency and using printed circuit board technology, as a starting point, we considered the simplest type of antennas, namely, printed planar dipoles. Such dipoles are appreciably larger than metasurface elements used in our earlier works and thus are expected to have equally appreciable higher power density at their input terminals. Since our interest is in large arrays (as opposed to powering electrically small devices such as medical implants), when the printed dipoles were placed in an array form, it was found that if the dipoles were fed off center and placed asymmetrically and covered with a dielectric material, a significant enhancement in the received power is observed. Specifically, it was found that the dielectric layer must be placed in the direction of the incoming wave to affect the surface impedance seen by the incoming waves. We emphasize that our findings are empirical. The evolution in these findings led to the unit cell structure shown in Fig. 1. The unit cell consists of two asymmetrical dipolar metallic elements covered with a high-permittivity dielectric superstrate. To achieve full energy absorption at the chosen frequency of 3.4 GHz, the harvester’s unit cell occupied a footprint of 18.7 mm × 38.4 mm. (We note that there was no particular reason for choosing this frequency aside from compatibility with our limited testing setup and facilities.) The covering dielectric superstrate was Rogers TMM-10i material of a thickness of t = 6.35 mm having a dielectric constant of εr = 9.8 and a loss tangent of 0.002. The dielectric superstrate was chosen for its low loss such that the absorbed energy was not dissipated within the dielectric material. The asymmetrical dipoles have a length of L = 14.7 mm, width of W = 6.80 mm, gap of g = 1 mm, and spacing between adjacent unit cells of sx = 1 mm and s y = 5 mm. The separation between the two dipoles is d = 14.8 mm. The two asymmetrical dipolar elements were initially terminated by 50- resistive loads across their gaps as illustrated in Fig. 1. The input impedance of the unit cell as seen from the gap can be controlled by varying the separation distance d between the two dipolar elements. This design feature is important for dc conversion when a diode is placed across the feeding gap.

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Fig. 1. Schematic showing the unit cell of the proposed dipole absorber covered with a high-permittivity dielectric superstrate.

Later, we show that the input impedance of the dipoles can be tailored by tuning d in order to match a diode that is needed for eventual dc rectification. In fact, such tuning eliminates the use of a matching network between the antenna and the diode. All the numerical simulations needed for this paper used ANSYS HFSS [28]. Since numerous simulations were required during our design development stage, for computational efficiency, a single unit cell was simulated on the assumption that it lies in an infinite periodic structure. Since an infinite array is neither practical nor the intended goal of this paper, a unit cell positioned in the middle of a large array is expected to have highly similar behavior to a unit cell placed in an infinite array. To this end, the periodic boundary condition was used to model a periodically infinite structure in the x y plane of the unit cell [29]. Master and slave boundaries were applied on the faces parallel to the wave propagation direction (−ˆz direction). The unit cell of the energy harvesting surface was illuminated by a normally incident plane wave propagating in the −ˆz direction using a Floquet port. Floquet excitation modes were used to simulate the incident wave with specular modes consisting of two orthogonally polarized plane waves propagating normally to the x y plane of the unit cell. Initially, and for the purpose of gauging the effectiveness of the absorber to absorb and channel energy to a load, lumped resistors were placed across the gaps of the two metallic elements. These resistive loads will be replaced later by a rectification circuit having an input impedance matched to the unit cell’s impedance at the maximum power absorption frequency. The radiation to ac/dc absorption efficiency describes the ability of an absorber to capture the energy per footprint area [9]. What we refer to in this paper as a radiation to ac efficiency is the efficiency of the harvester to transfer the total power incident on a specific area to available power at the feed (i.e., where a resistive load or rectifying circuitry is placed, referred to as the load location in Fig. 1). The radiation to ac conversion efficiency is calculated by calculating the footprint (surface area) in square meters. The radiation to ac conversion efficiency of an energy harvester occupying a specific footprint is then described as ηRad−ac = Pout /Pin

(1)

where Pin is the total time-average power incident on the footprint, and Pout is the available time-average ac power received by the harvesters’ collectors (i.e., where a resistive

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Fig. 3. Schematic showing the unit cell of the proposed energy harvesting surface showing the connectors used for dc channeling.

Fig. 2. Simulated radiation to ac conversion efficiency for the unit cell when the structure is illuminated by a plane wave traveling in the −ˆz direction. The dipoles are terminated with 50- loads.

load or a diode rectifying circuity is placed) and is given by Pout =

N 

Vi2 /Ri

(2)

n=1

where Vi is the voltage across the resistance of the ith collector, Ri , and N is the total number of collectors. This radiation to ac efficiency definition is indicative of the ability of the harvester to utilize the available electromagnetic energy incident on a given area and the ability of the harvester or its collectors to deliver the absorbed energy to resistive loads. This definition provides a measure of how efficient the absorber, occupying a specific footprint, in converting the available incident microwave power to RF energy. The simulated radiation to ac power conversion efficiency of the surface (assuming infinite periodicity) is shown in Fig. 2. Based purely on empirical findings, the asymmetrical dipoles were found to give higher harvesting efficiency than symmetrical dipoles with center feeding or off-center feeding. In fact, the asymmetrical case provided more than 40% increase in efficiency compared with the other cases. The design of the unit cell presented shows a very high radiation to ac conversion efficiency approaching unity at a frequency very close to the design frequency of 3.4 GHz. If harvesting at a different frequency is desired, the design parameters such as the dipoles length and width, superstrate material and thickness, and separation distance between the dipole can all be adjusted accordingly. It is important to realize that the near-unity radiation to ac conversion efficiency achieved here used energy harvesting elements having dimensions appreciably larger than the metamaterial elements used in earlier works [9], [12]. The significance of such an achievement is primarily due to reducing the requirement for the number of rectification diodes per footprint, which implies higher power density per diode. This in turn is expected to maximize diode turn-ON time and thus higher overall efficiency in practical applications where the incident field power density cannot be predicted in advance. For radiation to dc conversion, the input impedances of the dipoles need to be analyzed. Lumped excitation ports were placed at the asymmetrical dipolar elements’ terminals (i.e., replacing the lumped loads used to determine the

radiation to ac conversion efficiency) in order to calculate the input impedances of the dipoles. The Floquet port in this case was replaced by a radiation boundary on top of the structure. For a fixed size unit cell of the proposed absorber, the input impedance appeared at the dipolar elements’ terminals can be controlled by changing the interspacing between the asymmetrical dipoles. An HSMS-2860 Schottky diode was considered in this paper. The diode has an input impedance of Z d = 184 − j 45  when terminated with a load of 300  at a frequency of 3.4 GHz (the input impedance was obtained from its model [30].) DC channeling can be accomplished by connecting a diode directly across each dipole terminals and routing the resulting dc currents through copper planar wires placed on the same plane of the dipoles or through vias to a different layer. The first option introduces additional wiring that can potentially alter the absorption effectiveness of the unit cell, whereas the latter option requires the fabrication of vias and an additional metallization layer, both of which add appreciable fabrication cost. Instead, we introduced small connectors to the unit cell as shown in Fig. 3. Naturally, the dc connectors altered the topology of the unit cell and the dipoles; however, their input impedance was slightly affected. In fact, the dc connectors essentially increased the size of the dipoles. Thus, to maintain maximum absorption around the frequency of interest (3.4 GHz), the unit cell dimensions and the spacing between the asymmetrical dipolar elements were modified to achieve the desired input impedance, which is the conjugate of the diode’s impedance. The dimensions of the modified unit cell became the following: L = 8.35 mm, W = 6.80 mm, v = 2 mm, g = 1 mm, d = 10.79 mm, sx = 1 mm, and s y = 7 mm. The size of the unit cell became 12.35 mm × 38.39 mm. The thickness of the superstrate was kept unchanged at t = 6.35 mm. To verify the surface performance of the modified design (with dc connectors), the unit cell dipoles were terminated by lumped elements having the conjugate impedance of the dipoles. The radiation to ac power conversion efficiency for the unit cell with dc connectors is shown in Fig. 4 when the absorber was illuminated by a plane wave traveling in the −ˆz direction. In some applications, the harvesting surface is placed above a conducting surface. Therefore, we tested the proposed surface when backed by a perfect electrically conducting (PEC) surface. A low-permittivity substrate separated a PEC surface from the metalized layer of the harvester as shown

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Fig. 4. Simulated radiation to ac power conversion efficiency for the modified design with dc connectors (infinite periodic array) when illuminated by a plane wave traveling in the −ˆz direction. The dipoles were terminated with 184–j45 .

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Fig. 7. Schematic showing the proposed planar dipole array energy absorber covered with a dielectric superstrate under illumination by a transmitting horn antenna.

Fig. 5. Schematic of the side view for the dielectric and PEC backed harvester.

Fig. 8. Simulated radiation to ac absorption for the finite array when the structure is illuminated by a plane wave traveling in the −ˆz direction.

ac efficiency for the finite array is shown in Fig. 8 when it was illuminated by a plane wave traveling in the −ˆz direction. The total available absorbed power is given by Fig. 6. Simulated radiation to ac power conversion efficiency for the finite array when backed by a low-permittivity dielectric layer and PEC plane.

in Fig. 5. Two different grounded substrate thicknesses of t1 = 1.575 and 3.175 mm of Rogers 5880 material having a dielectric constant of εr = 2.2 were tested. Fig. 6 shows numerical simulation of the radiation to ac efficiency. The grounded low-permittivity substrate is observed to provide a wider frequency absorption range. Next, we analyze the behavior of a finite array. The numerical setup is shown in Fig. 7 where a horn antenna is used in the simulation to provide a physically realistic plane wave as opposed to the plane wave feature available in HFSS. A finite size array consisting of 9 × 3 unit cells in the x y plane using a Rogers TMM-10i superstrate was numerically simulated. The dipoles were all terminated by their conjugate input impedance (recall that the unit cell was designed such that the input impedances of the dipoles were the conjugate of the input impedance of the Schottky diode at 3.4 GHz and when terminated with 300 ). The entire array size was 111.2 mm × 115.2 mm. The simulated radiation to

Pout =

N 

(n)

pout

(3)

n=1 (n) where N is the number of elements in the array and pout is the received power from each array element. Fig. 9 shows the simulated surface current on the absorber’s conducting elements and the magnitude of the electric field on the absorber’s bottom layer. Both the current and electric field were taken at the maximum absorption frequency 3.4 GHz with incidence plane wave propagating in the −ˆz direction. Of course, since the array is not infinite, we observed some nonuniformity in the current and field distributions. Interestingly, however, we observed significantly higher current density through the dc connectors in comparison with the impedance loads. To visualize the physical response of the absorbers to the plane wave excitation, the time-averaged Poynting vector of the total field is shown in Fig. 10 at 3.4 GHz when the structure is illuminated by a plane wave traveling in the −ˆz direction. Fig. 10 clearly shows the channeling of the power of the incident plane wave as it propagates closer to the surface

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Fig. 11. Fabricated bottom view of the planar dipole absorbers covered by high-permittivity superstrate made of Rogers TMM-10i material.

Fig. 9. (a) Surface current distribution on the dipole elements and (b) electric field magnitude on the bottom layer at the maximum absorption frequency of 3.4 GHz when illuminated by an plane wave incident in the −ˆz direction. The surface current highest intensity (red) corresponds to 7 × 10−2 A/m and the lowest intensity (blue) corresponds to 8 × 10−5 A/m. The electric field highest intensity (red) corresponds to 12 V/m and the lowest intensity (blue) corresponds to 1.25 × 10−2 V/m.

of 9 × 3 made with the TMM-10i superstrate. The surface was illuminated by a broadband commercially available horn antenna (0.7–18-GHz frequency range with a maximum gain of 14.71 dBi) placed at a distance of 1 m for a plane wave illumination. The broadband horn antenna was excited by a Keysight signal generator through with a 42-dB gain power amplifier to investigate the required power level that would allow the diode to operate at its maximum efficiency. The experiments were carried out in an anechoic chamber for maximum accuracy. Also, the impact of the horn antenna gain level over the testing frequency was carefully considered in calculating the power provided by the horn antenna. The cable loss connecting the horn antenna to the signal generator was also measured over the testing frequency. The following formula was then employed to calculate the power provided by the horn antenna at the absorber plane Pin : Pin = A × Pt

(4)

where P G t ( f ). (5) 4π R 2 A is the harvester surface area (footprint), Pt is the radiated power density generated by the horn at the harvester surface plane in the absence of the harvester, and P is the power fed to the transmitting horn’s input terminal. G t ( f ) is the gain of the horn antenna as a function of frequency and R is the distance between the horn antenna and the array. The available power at the absorber plane, Pin , was then calculated from which the overall power absorption of the array can be obtained using (1). The nonlinear performance of the rectifying diodes will have an effect on the absorber rectifying/absorption performance for various dc connections topologies (i.e., series or parallel connection of the absorber’s elements), loads, incident power, and frequencies [31], [32]. Therefore, a balanced topology between series and parallel connections was considered for the dc measurements where each two columns of the array elements in the x-axis ˆ were connected in series and then connected in parallel. A schematic representing the chosen dc channeling topology is shown in Fig. 12. The fabricated planar dipole array using Rogers TMM-10i shown in Fig. 11 was illuminated by the horn antenna in the −ˆz direction (refer to Fig. 7 for the illumination directions) where the array absorber was experientially tested in an anechoic chamber as shown in Fig. 13. The load impedance Pt =

Fig. 10. Poynting vector of the total incident field at a maximum absorption frequency of 3.4 GHz when the structure is illuminated by an incident plane wave traveling in the −ˆz direction. The highest intensity (red) corresponds to 2×10−3 W/m2 and the lowest intensity (blue) corresponds to 8×10−5 W/m2 .

and eventually into the impedance loads. Note that despite the absorber surface not being homogeneous, not in the macroscopic sense but rather not being composed of periodic arrangement of electrically small cells, as in metasruface type absorbers, practically all the energy is channeled into the loads. The significance of this physical insight is that one does not need an absorbing surface with uniform surface impedance matched to free space to achieve full energy absorption. III. E XPERIMENTAL V ERIFICATION The planar array absorber was fabricated using the above simulated array’s specifications. Fig. 11 shows an array

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Fig. 12. Schematic showing the dc current channeling configuration adopted for the harvesting array.

Fig. 14. Measurement dc absorption as a function of (a) load impedance and (b) available power at the absorber plane Pin when the structure is illuminated by a horn antenna in the −ˆz direction. Fig. 13. Absorber under illumination by a broadband horn antenna for dc measurements.

and power at the receiving end were swept at the highest absorption frequency of 3.4 GHz. The purpose of those sweeps was to find the optimal dc load and input power that will deliver maximum power across the load. Fig. 14 shows that the harvesting surface achieves maximum power across the load when illuminated by a power level of 27 mW (i.e., the available power at the absorber plane, Pin ) with a load of 2 k for −ˆz illumination. This optimal load value was in fact expected, which is the result of connecting 18 dipolar elements (two columns of the array elements along the x-axis) ˆ of the harvesting array with 300- termination loads in series and then connected in parallel as illustrated in Fig. 12. The dc absorption efficiency for the illumination in the −ˆz direction was then recorded at multiple frequency points at which the absorber had maximum power across the load (i.e., with the optimal obtained input power and load). This is given in Fig. 15. The absorber achieved a radiation to dc conversion efficiency of 76%. Finally, the absorber was tested for capturing the incoming radiation at different angles of incidence. Fig. 16 shows the measurement results for different angels of illumination. The simulated radiation to dc absorption ηRad−dc efficiency given in Figs. 15 and 16 was calculated as follows: ηRad−dc = ηRad−ac × ηac−dc (6)

Fig. 15. Efficiency of the measured and simulated radiation to dc absorption of the dipole array harvester as a function of frequency when the structure is illuminated by a horn antenna in the −ˆz direction.

where ηRad−ac is the harvester radiation to ac efficiency and ηac−dc is efficiency of the rectifier over the tested frequency range 2–4 GHz, with a 300- termination load. The discrepancy between the simulated and measured results shown in Figs. 15 and 16 is mainly due to the nonlinearity of the diode. Given that the diode’s turn-ON voltages are intrinsically dependent on the operating frequency, level of power, and the load connected to the circuit, the diode’s input impedance was determined only at 3.4 GHz while terminated with a 300- resistor such that the energy harvesting structure was optimally designed to be matched with the diodes only at 3.4 GHz with a load termination of 300 . Therefore,

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ACKNOWLEDGMENT The authors would like to thank the Saudi Arabian Ministry of Education, Prince Sattam University, and the Natural Sciences and Engineering Research Council, Canada, for the support. R EFERENCES

Fig. 16. Efficiency of the measured and simulated radiation to dc absorption of the dipole array harvester as a function of different incident illuminations when E-field polarized in the (+ x) ˆ direction. (Note: the normal incident illumination is in the (−ˆz ) direction.)

mismatching between the diode impedance and the array elements’ impedance is expected over frequencies different from 3.4 GHz. Consequently, determining the behavior of the harvester outside a narrow range of frequencies around 3.4 GHz can only be accurately determined via measurements as simulation provides only a crude approximation outside this narrow range. IV. C ONCLUSION We have presented a surface for electromagnetic energy harvesting using unit cells consisting of two asymmetrical dipolar elements covered with a high-permittivity dielectric superstrate. The energy harvesting surface was tested for its potential to absorb and channel energy by placing a resistive load across the gaps of the dipoles. The dimensions of the dipoles were optimized in order to provide maximum power transfer to a rectification diode. The design essentially evolved from a simple printed center-fed dipole without any dielectric material to off-center fed dipoles with a dielectric layer placed in the direction of the incoming energy. We introduced conducting elements to effectively turn the planar dipoles themselves into a dc channeling network. An overall radiation to dc power conversion efficiency of 76% was obtained experimentally for normal incidence. We note that, to the best of our knowledge, the radiation to dc conversion efficiency achieved by our energy harvesting surface is higher than what was achieved using all previous surfaces reported in the literature including metasurfaces and planar antenna arrays [15], [16]. This paper showed that near-unity absorption of incoming electromagnetic energy can be made possible without the use of metasurfaces. In metasurface technology, the surface impedance of the metasurface is designed to be equal to that of free space. The surface proposed here achieved near-unity absorption (for ac) without the need to meet such a constraint. Unlike metasurface technology, the advantage of having fewer absorbing elements is to maximized energy density per diode, which in turn maximizes diode turn-ON time and overall efficiency.

[1] W. C. Brown, “The history of power transmission by radio waves,” IEEE Trans. Microw. Theory Techn., vol. MTT-32, no. 9, pp. 1230–1242, Sep. 1984. [2] P. E. Glaser, “An overview of the solar power satellite option,” IEEE Trans. Microw. Theory Techn., vol. 40, no. 6, pp. 1230–1238, Jun. 1992. [3] C. R. Valenta and G. D. Durgin, “Harvesting wireless power: Survey of energy-harvester conversion efficiency in far-field, wireless power transfer systems,” IEEE Microw. Mag., vol. 15, no. 4, pp. 108–120, Jun. 2014. [4] B. Strassner and K. Chang, “Microwave power transmission: Historical milestones and system components,” Proc. IEEE, vol. 101, no. 6, pp. 1379–1396, Jun. 2013. [5] A. Harb, “Energy harvesting: State-of-the-art,” Renew. Energy, vol. 36, no. 10, pp. 2641–2654, 2011. [6] J. A. Paradiso and T. Starner, “Energy scavenging for mobile and wireless electronics,” IEEE Pervasive Comput., vol. 4, no. 1, pp. 18–27, Jan./Mar. 2005. [7] L. Ukkonen, L. Sydanheimo, and M. Kivikoski, “Effects of metallic plate size on the performance of microstrip patch-type tag antennas for passive RFID,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 410–413, 2005. [8] H. Jabbar, Y. S. Song, and T. T. Jeong, “RF energy harvesting system and circuits for charging of mobile devices,” IEEE Trans. Consum. Electron., vol. 56, no. 1, pp. 247–253, Feb. 2010. [9] O. M. Ramahi, T. S. Almoneef, M. AlShareef, and M. S. Boybay, “Metamaterial particles for electromagnetic energy harvesting,” Appl. Phys. Lett., vol. 101, no. 17, 2012, Art. no. 173903. [10] B. Alavikia, T. S. Almoneef, and O. M. Ramahi, “Electromagnetic energy harvesting using complementary split-ring resonators,” Appl. Phys. Lett., vol. 104, no. 16, 2014, Art. no. 163903. [11] M. R. AlShareef and O. M. Ramahi, “Electrically small particles combining even- and odd-mode currents for microwave energy harvesting,” Appl. Phys. Lett., vol. 104, no. 25, 2014, Art. no. 253906. [12] T. S. Almoneef and O. M. Ramahi, “Metamaterial electromagnetic energy harvester with near unity efficiency,” Appl. Phys. Lett., vol. 106, no. 15, 2015, Art. no. 153902. [13] M. El Badawe and O. M. Ramahi, “Polarization independent metasurface energy harvester,” in Proc. 17th Annu. IEEE Wireless Microw. Technol. Conf., Apr. 2016, pp. 1–3. [14] M. El Badawe, T. S. Almoneef, and O. M. Ramahi, “A metasurface for conversion of electromagnetic radiation to DC,” AIP Adv., vol. 7, no. 3, 2017, Art. no. 035112. [15] F. Erkmen, T. S. Almoneef, and O. M. Ramahi, “Electromagnetic energy harvesting using full-wave rectification,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 5, pp. 1843–1851, May 2017. [16] P. Xu, S.-Y. Wang, and W. Geyi, “Design of an effective energy receiving adapter for microwave wireless power transmission application,” AIP Adv., vol. 6, no. 10, 2016, Art. no. 105010. [17] E. Falkenstein, M. Roberg, and Z. Popovic, “Low-power wireless power delivery,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 7, pp. 2277–2286, Jul. 2012. [18] M. Arrawatia, M. S. Baghini, and G. Kumar, “Differential microstrip antenna for RF energy harvesting,” IEEE Trans. Antennas Propag., vol. 63, no. 4, pp. 1581–1588, Apr. 2015. [19] S. Ladan, N. Ghassemi, A. Ghiotto, and K. Wu, “Highly efficient compact rectenna for wireless energy harvesting application,” IEEE Microw. Mag., vol. 14, no. 1, pp. 117–122, Jan. 2013. [20] C. Song, Y. Huang, J. Zhou, J. Zhang, S. Yuan, and P. Carter, “A high-efficiency broadband rectenna for ambient wireless energy harvesting,” IEEE Trans. Antennas Propag., vol. 63, no. 8, pp. 3486–3495, Aug. 2015. [21] U. Olgun, C.-C. Chen, and J. L. Volakis, “Investigation of rectenna array configurations for enhanced RF power harvesting,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 262–265, 2011. [22] Z. Popovic et al., “Scalable RF energy harvesting,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 1046–1056, Apr. 2014.

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[23] F. Xie, G. M. Yang, and W. Geyi, “Optimal design of an antenna array for energy harvesting,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 155–158, 2013. [24] J. Zbitou, M. Latrach, and S. Toutain, “Hybrid rectenna and monolithic integrated zero-bias microwave rectifier,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 1, pp. 147–152, Jan. 2006. [25] J. A. Hagerty, F. B. Helmbrecht, W. H. McCalpin, R. Zane, and Z. B. Popovic, “Recycling ambient microwave energy with broad-band rectenna arrays,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 3, pp. 1014–1024, Mar. 2004. [26] H. Sun, Y.-X. Guo, M. He, and Z. Zhong, “A dual-band rectenna using broadband Yagi antenna array for ambient RF power harvesting,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 918–921, 2013. [27] Y.-J. Ren and K. Chang, “5.8-GHz circularly polarized dual-diode rectenna and rectenna array for microwave power transmission,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 4, pp. 1495–1502, Jun. 2006. [28] ANSYS HFSS Version 16.0.0, Ansys Inc. Accessed: Jul. 18, 2017. [Online]. Available: http://www.ansys.com [29] ANSYS HFSS. (1970). Workshop 9–1: Unit Cell Analysis (Infinite Array). Accessed: Jul. 18, 2017. [Online]. Available: https://www.scribd. com/document/320371134/Ansys-Hfss-Antenna-w09-1-Unit-Cell [30] J. O. McSpadden, L. Fan, and K. Chang, “Design and experiments of a high-conversion-efficiency 5.8-GHz rectenna,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 12, pp. 2053–2060, Dec. 1998. [31] R. J. Gutmann and J. M. Borrego, “Power combining in an array of microwave power rectifiers,” IEEE Trans. Microw. Theory Techn., vol. MTT-27, no. 12, pp. 958–968, Dec. 1979. [32] N. Shinohara and H. Matsumoto, “Dependence of dc output of a rectenna array on the method of interconnection of its array elements,” Elect. Eng. Jpn., vol. 125, no. 1, pp. 9–17, 1998. Ahmed Z. Ashoor (GS’12–M’12) was born in Ras Tanura, Saudi Arabia, in 1981. He received the B.S. degree in physics from King Saud University, Riyadh, Saudi Arabia, in 2002, and the M.A.Sc. degree in electronics and computer engineering from the University of Waterloo, Waterloo, ON, Canada, in 2012, where he is currently pursuing the Ph.D. degree in electrical and computer engineering. He was an Intern with the Inspection Department/Nondestructive Testing, Saudi Aramco Oil Company, Dhahran, Saudi Arabia. In 2003, he joined the Al-Ahsa College of Technology, al-Mubarraz, Saudi Arabia, and Al-Dammam College of Technology, Dammam, Saudi Arabia, as a Trainer/Instructor, where he taught a first-year physics course. In 2004, he joined industry as an NDT Engineer and Radiation Safety Officer (RSO), where he was an RSO supporting various inspection activities on different projects at the Saudi ARAMCO facilities. His current research interests include radiographic, ultrasound, electromagnetic, visual, eddy current, magnetic flux leakage, infrared testing, electromagnetic energy harvesting, wireless power transfer, and nanoantenna and metamaterial.

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Thamer S. Almoneef (GS’10–M’12) received the B.S. degree in electrical and computer engineering from Dalhousie University, Halifax, NS, Canada, in 2009, and the M.A.Sc. and Ph.D. degrees in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada, in 2012 and 2017, respectively. In 2012, he joined Prince Sattam University, Al-Kharj, Saudi Arabia, as a Lecturer, where he is currently an Assistant Professor with the Department of Electrical and Computer Engineering. He has authored or co-authored more than 20 refereed journals and conference papers. His current research interests include antenna theory, metamaterials and its wide range applications, metamaterial absorbers, electrically small resonators, rectennas, electromagnetic energy harvesting, and renewable energy. Dr. Almoneef was the recipient of a Scholarship from Prince Sattam University for his Ph.D. study.

Omar M. Ramahi (F’09) was born in Jerusalem, Palestine. He received the B.S. (Highest Hons.) degrees in mathematics and electrical and computer engineering from Oregon State University, Corvallis, OR, USA, and the Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana–Champaign, Champaign, IL, USA. He was with Digital Equipment Corporation (presently, HP), Maynard, MA, USA, where he was a member of the Alpha Server Product Development Group. In 2000, he joined the faculty of the James Clark School of Engineering, University of Maryland at College Park, College Park, MD, USA, as an Assistant Professor and later as a Tenured Associate Professor. He was a faculty member with the CALCE Electronic Products and Systems Center, College Park, MD, USA. He is currently a Professor with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada. He has authored or co-authored over 380 journal and conference technical papers on topics related to the electromagnetic phenomena and computational techniques to understand the same. He coauthored EMI/EMC Computational Modeling Handbook [Kluwer, 1998 (1st edition), Springer-Verlag, 2001 (2nd edition), 2005 (Japan edition)]. Prof. Ramahi was a recipient of the 2004 University of Maryland Pi Tau Sigma Purple Cam Shaft Award. He was the recipient of the Excellent Paper Award of the 2004 International Symposium on Electromagnetic Compatibility, Sendai, Japan, and the 2010 University of Waterloo Award for Excellence in Graduate Supervision. In 2012, he was a recipient of the IEEE Electromagnetic Compatibility Society Technical Achievement Award.

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A 40-nm CMOS Complex Permittivity Sensing Pixel for Material Characterization at Microwave Frequencies Gerasimos Vlachogiannakis , Student Member, IEEE, Michiel A. P. Pertijs, Senior Member, IEEE, Marco Spirito, Member, IEEE, and Leo C. N. de Vreede, Senior Member, IEEE Abstract— A compact sensing pixel for the determination of the localized complex permittivity at microwave frequencies is proposed. Implemented in the 40-nm CMOS, the architecture comprises a square patch, interfaced to the material-undertest sample, that provides permittivity-dependent admittance. The patch admittance is read out by embedding the patch in a double-balanced, RF-driven Wheatstone bridge. The bridge is cascaded by a linear, low-intermediate frequency switching downconversion mixer, and is driven by a square wave that allows simultaneous characterization of multiple harmonics, thus increasing measurement speed and extending the frequency range of operation. In order to allow complex permittivity measurement, a calibration procedure has been developed for the sensor. Measurement results of liquids show good agreement with theoretical values, and the measured relative permittivity resolution is better than 0.3 over a 0.1–10-GHz range. The proposed implementation features a measurement speed of 1 ms and occupies an active area of 0.15×0.3 mm2 , allowing for future compact arrays of multiple sensors that facilitate 2-D dielectric imaging based on permittivity contrast. Index Terms— Biomedical sensors, bridge circuits, complex permittivity measurement, integrated microwave circuits, microwave sensors.

I. I NTRODUCTION

B

ROADBAND dielectric spectroscopy at microwave frequencies has been identified as a promising tool for a large number of applications, ranging from the agricultural, food, and automotive industry to the biomedical domain [1]–[8]. This method relies on the fact that the dielectric footprint of various materials of interest, i.e., their complex permittivity across frequency, varies in conjunction with a parameter that needs to be detected or quantified. To highlight a few examples, in agriculture, the complex permittivity of fruits and vegetables has been correlated with changes in temperature, water, and inorganic material content [1]–[3], while in the automotive industry, it is the preferred method for oil and fuel quality inspection [4], [5]. On the biomedical side of the application spectrum, examples include

Manuscript received March 15, 2017; revised August 1, 2017; accepted September 2, 2017. This work was supported by the Dutch Technology Foundation (STW/NWO) under Project INFORMER 13010. (Corresponding author: Gerasimos Vlachogiannakis.) The authors are with the Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628CD Delft, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2753228

blood glucose monitoring [6] and ex vivo or in vivo cancer detection and assessment [8], [9]. The latter application is supported by measurements on bulk animal and human tissue, suggesting that the permittivity of cancer tissue can vary by up to 20% compared with healthy tissue [10], [11]. Despite the promising potential suggested by these studies, conventional microwave permittivity measurement techniques, used to acquire the aforementioned literature data, employ expensive and bulky equipment, such as vector network analyzers (VNAs) and probe or cavity sensors interfaced to the material under test (MUT) [12], [13]. These setups are not suitable for most practical application scenarios, such as outdoor, remote-location measurements, and point-of-care medical testing. Moreover, their high cost hinders potential wider adoption of the technology. Miniaturization of sensors and measurement systems is, therefore, essential in order to leverage the true potential of microwave permittivity sensing in real-life applications. Moreover, miniaturized sensors can facilitate new applications that deviate from the bulk-level measurement regime, such as the unexplored area of 2-D sensor arrays for permittivity contrast measurement and visualization at microwave frequencies. Such imaging functionality can prove useful in a variety of applications such as the following: 1) label-free, in vivo cancer visualization as an assisting tool in removal surgery [14]; 2) food and flower quality inspection for early detection of storage disorders (e.g., browning, skin spots, and so on); 3) evaluation of drug penetration through the skin; 4) nondestructive film coating testing in industry. A differentiation should be made at this point between microwave permittivity sensors and low-frequency permittivity/impedance sensors, operating below 100 MHz. For the latter, arrayed implementations have already been implemented successfully [15], [16]. Nevertheless, motivation to move toward broadband microwave frequency implementations still exists for two main reasons: 1) in order to achieve better penetration in the MUT and 2) to employ the higher redundancy implicit in acquiring a permittivity data set, which is more complete and flexible in the frequency domain. Such redundancy is directly linked to increased sensitivity and specificity in biomedical applications. To enable such imaging systems, focus has to be put on a fast readout, with acceptable resolution to fulfill the application requirement, as well as the overall size of the sensor and

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its signal conditioning circuitry, since this will determine its scalability in a dense array toward a fine spatial resolution. Efforts toward miniaturization of microwave permittivity sensing systems have been mainly concentrated toward CMOS implementations because of the ultimate form factor that CMOS offers. Several microwave CMOS implementations during the last years have demonstrated accurate permittivity readout [17]–[24]. Oscillator-based approaches exist, which are very narrowband, area-consuming, and limited to measurement of the real part of permittivity, and thus are not suitable for the implementation of a broadband permittivity sensing pixel [17], [19], [22], [24]. Several other implementations achieve an operation frequency range of at least a decade by employing broadband downconversion [18], [20], [21], [23] or wide-band PLL-based architectures [22]. However, since they are not meant for imaging applications, little optimization and analysis have been done on the readout speed, resulting in potentially long measurement times. Moreover, the active area still remains quite large if implementation of a sub-mmresolution imager is targeted. In the following sections, we detail an integrated complex permittivity sensor, suitable for use as an imaging pixel, which was prototyped in 40-nm CMOS and occupies sub-mm2 area while achieving fast readout. The proposed sensor, briefly presented in [25], features a single-ended patch sensing element, embedded in a fully differential double-balanced RF-driven impedance bridge. A multiharmonic measurement scheme is employed to extend the frequency range and increase the effective measurement speed. In this paper, we analyze the utilized sensing element in depth, and develop a calibration procedure, based on the analysis of the RF bridge. Moreover, the noise sources that contribute to the system resolution limit are identified and their contribution is quantified. Additional measurement data are complementing the preliminary results reported in [25] that demonstrated the ability to measure material complex permittivity. Independent measurements with the sensing pixel loaded by a probe that offers a known termination are used to validate the bridge transfer characteristic, while the statistical data of material measurements have been collected to evaluate the permittivity resolution of the sensor when the fundamental, third, and fifth harmonic are measured. This paper is organized as follows. Section II analyzes the basic principles behind the system architecture, including the near-field patch sensor, the RF impedance bridge it is embedded in, as well as the multiharmonic intermediate frequency (IF) downconversion readout concept. Section III describes the physical implementation of the permittivitysensing system in a 40-nm CMOS technology. In Section IV, a calibration procedure for the developed sensor is given, and the resulting accuracy and resolution are discussed. Experimental results are presented in Section V. Finally, the conclusions are drawn in Section VI. II. S YSTEM A RCHITECTURE To address the aforementioned application scenarios, it is desirable that the sensor features the following qualities: 1) broadband operation that allows flexibility in choice of frequency;

Fig. 1. 3-D depiction of (a) differential and (b) single-ended patch sensing element.

2) complex material permittivity detection, i.e., ability to detect both real and imaginary part of the permittivity; 3) suitability for embedding in a 2-D array for permittivity contrast imaging, implying small size and fast readout. The proposed architecture consists of a near-field patch sensor, an RF-driven impedance bridge in a double-balanced configuration, and a multiharmonic, IF downconversion scheme. A. Near-Field Sensor The sensing element translates the relative permittivity of the material, expressed as a frequency-dependent complex number   (ω) =   (ω) − j   (ω), into a lumped equivalent complex admittance that can be read out by subsequent circuitry. Previously reported CMOS permittivity sensors typically employ differential capacitive sensing elements, similar to the one shown in Fig. 1(a), implemented on the top metal of the CMOS metal stack, with a passivation opening for direct contact to the MUT [17], [19]–[22]. These sensor types provide convenient access to both terminals [P+ and P− in Fig. 1(a)] and are directly compatible with fully differential readout chains. However, due to their planar configuration, the electric field is mainly concentrated in the vicinity of the sensor surface, i.e., the surface-MUT interface. On the contrary, the electric field lines of a single-ended metal patch sensor, portrayed in Fig. 1(b), penetrate deeper in the MUT, thus allowing sensing further from the sensor-MUT interface. To demonstrate this, EM simulations were carried out to determine the electric field as a function of vertical distance from the sensor surface, using a commercial 3-D EM simulation tool (Keysight EMPro). The two simulated sensors occupy an area of 100 × 100 μm2 , and a distance of 10 μm between fingers was chosen for the differential sensor. A typical 40-nm CMOS metal stack was considered, and the EM simulation was carried out at 1 GHz, in a worst case scenario where the sensor is interfaced to air (  = 1 − j 0). As seen in the simulation results in Fig. 2, a much steeper decay of the electric field is evident in the case of the differential sensor. At a distance of 300 μm, the electric field magnitude is approximately 100 dB lower than the maximum strength, whereas for the patch sensor, this reduction is in the order of 70 dB, a difference of 30 dB. A patch sensor is, therefore, less sensitive to potential air gaps, since a smaller portion of the field is concentrated at the interface. This property is desired in solid or semirigid

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TABLE I PARAMETERS OF L INEAR - TO -Y M ODEL

Fig. 2. EM simulation of normalized electric field magnitude versus vertical distance from the sensor interface for two types of sensors both occupying the same 100 × 100 μm2 area: a single-ended patch and a differential capacitor with 10 μm between fingers.

expected to represent a lossy capacitor of which the reactive and resistive behavior will strongly depend on the real and imaginary parts of the MUT permittivity, respectively. Hence, the admittance Y P at the patch node can be expressed as a parallel combination of a material-dependent admittance YMUT ≈ G MUT (  ) + j ωCMUT (  ) and the baseline admittance Y0 = j ωC0 , yielding Y P = Y0 + YMUT . In order to quantify the permittivity-to-admittance behavior of the patch, a 3-D model of a 100 × 100-μm2 patch on a realistic representation of the available 40-nm CMOS stack, in direct contact with an MUT, was simulated versus varying   and   . The solid lines in Fig. 3(b) show the capacitance and the conductance of node P versus   and   , for different values of   and   , respectively, at a simulation frequency of 1 GHz. An explicit relation of capacitance to   and conductance to   exists that can be linearly approximated by Y P (  ,   , ω) ≈ αi · ω ·   + j ω · (C0 + αr   )

Fig. 3. (a) Cross section of utilized patch sensing element and (b) equivalent patch capacitance and conductance from EM simulations (solid lines) and RFM model (dots) for various values of   and   at f = 1 GHz.

material measurements (e.g., biological tissue), but also in applications when a permittivity contrast measurement deeper in the MUT is targeted. Although the patch sensor is expected to provide a poorer isolation to neighboring pixels, the fact that it is not inherently bound to differential sensing also allows the use of more advanced driving schemes where multiple patches are used to inspect a sample. Examples of such schemes include multiphase patch excitation, selective differential sensing between different sensors, and bootstrapping, i.e., driving neighboring pixels without reading them in order to cancel capacitive cross coupling [26]. Based on the above, the patch configuration was favored as a sensing element in this implementation. Fig. 3(a) shows the cross section of the a square patch sensor implemented on the top metal of a generic CMOS stack. When the patch is in contact with air, the patch node P is loaded by the parallel-plate capacitance C0 , formed between the top metal and the ground plane. When interfaced to an MUT, the load will change depending on the MUT complex permittivity. Since permittivity relates to electric energy storage and loss (  and   , respectively), the sensing element is

(1)

where αr and αi are real parameters. Note that the ω contribution in the real part of the admittance results from the fact that the conductivity of the material is given by σ = ω  [27]. Table I summarizes the parameters of the model in (1) extracted after least square fitting with the EM-simulated curves. Although the linear model is simple, intuitive, and useful for preliminary analysis, it is clear from the simulated results of Fig. 3(b) that G MUT and CMUT also vary with   and   , respectively, an effect not captured by (1). For the purpose of calibration, a rational function model (RFM), fit from EM simulations, can be used to arrive to an analytical model, a methodology widely used in permittivity measurements performed with open-ended coaxial probes [13], [28], [29] √ p N  P n  n=1 p=1 αnp (  ) ( j ωa)  Y P ( , ω) ≈ j ωC0 + √  M Q 1 + m=1 q=1 βmq (   )q ( j ωa)m (2) where a is a scaling parameter, set equal to the patch dimension, and αnp and βmq are N × P and M × Q real model parameters, respectively. In order to find the parameters, (2) is fit with parametric EM simulations across   ,   , and frequency. A fit model with N = P = M = Q = 4 is deemed sufficient, since it already achieves 1% maximum deviation from simulations over a 0.1–10-GHz frequency range. B. RF Impedance Bridge Following the established analytical -to-Y model for the patch, a method of reading out the admittance is required.

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admittance, i.e., G L  G 0 and B L  B0 , (3) denotes that the output varies linearly with the measured load admittance YL . (4) 4Y0 This approximation, however, can result in large errors in the estimation of Y L . A more generic result that accounts for any value of measured load is possible, irrespective of how much it unbalances the bridge and without requiring any approximations. Indeed, assuming that Y L = 0, inverting (3) results in   1 4Y0 1 2+ . (5) = v b,o v in YL v b,o ≈ v in ·

Substituting for Y0 and Y L yields   1 1  = (2 + 4G 0 · G Lw + 4B0 · B Lw ) v b,o v in and

 

Fig. 4. Balanced impedance bridge, driven at RF frequency by a driver, with annotated signals and noise sources contributing to the total noise at the output of the bridge.

A Wheatstone bridge [30], [31] is a widely adopted method of measuring or sensing electrical impedance, since it offers a quantification of impedance variation relative to a constant baseline value, such as C0 in the case of the patch sensor. At RF frequencies, impedance bridges have been widely used in the broadband vector network analysis as directional detection elements, as an alternative or complementary to bidirectional couplers [32]. In this section, an alternative analysis of the ac-driven Wheatstone bridge with complex branch loads is presented. A mathematical manipulation of the bridge equation is performed to extract useful information for the calibration of the sensor. This analysis is later verified by the measurements of various known RF impedances in a probed measurement environment. Moreover, the bridge output noise is calculated to extract information about the minimum detection limit. 1) Bridge Analysis: Consider the RF impedance bridge shown in Fig. 4 with branch admittances Y0 and the load measurand Y L deviating from a baseline admittance Y0 . The bridge is excited at a given frequency ω with a signal of amplitude v in , through bridge driver that amplifies a signal v i of the same frequency. The differential output voltage of the bridge can be found after straightforward circuit analysis v b,o = v b,o+ − v b,o− = v in ·

YL 4Y0 + 2Y L

(3)

where Y L = G L + j B L and Y0 = G 0 + j B0 are the generic complex representation of the admittances. A common approximation is that, for small variations of the measured load

1 v b,o

 =

4 (B0 · G Lw − G 0 · B Lw ) v in

(6)

(7)

where G Lw : = G L /|Y L |2 and B Lw : = B L /|Y L |2 are defined as the weighted load conductance and susceptance values, respectively. Therefore, irrespective of deviation of Y L from Y0 , the real and imaginary parts of the inverse bridge differential output are the linear combinations of the weighted load conductance and susceptance. Formulating the bridge behavior as in (6) and (7) allows to present a linear relation between an output quantity (inverse of output) to the input quantity (weighted conductance and susceptance). In this manner, an intuitive calibration procedure can be obtained that is closer to the bridge operation, rather than utilizing the high-order polynomial fitting [18], [20], [21]. The calibration procedure is described in detail in Section IV-A. 2) Bridge Noise: In order to calculate the noise at the output of the bridge, we can break it down into three uncorrelated components shown in Fig. 4: thermal noise generated by the bridge resistive elements (v th,n ), flicker, shot, and thermal noise generated by any internal active elements driving the bridge (v dr,n ), and input noise to the bridge driver originating from the RF signal generator, either external or internal (v gen,n ). By applying superposition, the contribution of each component to the output noise can be analyzed. The total noise is thus the mean-square sum of these three components: 2 2 2 v n,o 2 = v th,n,bo + v dr,n,o + v gen,n,o . The thermal noise power at the differential output of the bridge is given by   ω+ω/2  1 2 v th,n,o = 4kT  · dω (8) 4Y0 + Y L ω−ω/2  ω+ω/2 4G 0 + G L · dω = 4kT 2 2 (4G + G 0 L ) + (4B0 + B L ) ω−ω/2 (9) where ω is the observation bandwidth. Since the complex permittivity is translated to conductance and capacitance, the bridge susceptance will essentially be that of a capacitance,

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Fig. 5. Thermal and external noise contributions to the bridge output noise versus bridge differential output voltage for different levels of IPN of the external source.

i.e., B = ωC. In addition, the observation bandwidth is typically much smaller than the frequency of interest (ω  ω), and thus, we can safely neglect the frequency variation of the integrated quantity   4G 0 + G L 2 v th,n,o ≈ 4kT ω. (10) (4G 0 + G L )2 + ω2 (4C0 + C L )2 As will be analyzed in Section III-C, a clipping buffer is used as the bridge driver. Assuming a quiet power supply, the contribution of noise from the bridge driver is in the form of cyclostationary phase-modulated (PM) noise that results from upconversion of thermal and flicker noise to the frequency of operation [33]. This noise will be scaled by the bridge similar to the bridge drive signal v in and can, therefore, be expressed as a function of the single-sideband (SSB) phase noise of the driver, L dr , and the differential output (v b,o ) of the bridge  ω 2 2 2 v dr,n,o =2 10 L dr (ω)/10 · v b,o · dω = IPNdr ·v b,o (11) 0

where IPNdr is the integrated phase noise (IPN) of the driver up to the measurement bandwidth ω. Similarly for the external generator noise, any amplitude-modulated component is suppressed by the buffer, but the PM noise will be propagated to the bridge through a phase noise transfer of unity, since any timing variation in the input of the switching buffer will be transferred directly to its output. As a consequence, the contribution of the generator noise to the output of the bridge can be expressed, identically to (11), as 2 2 v gen,n,o = IPNgen · v b,o

(12)

where IPNgen is the double sideband IPN of the generator within the measurement bandwidth ω. Notice from (11) and (12) that the noise components related to the bridge drive are proportional to the output power, which suggests that the more balanced the bridge is, the less the external noise contribution to the output. These contributions can be grouped together into what we can call external noise contributions. Fig. 5 shows how the two noise contributions (thermal and external) will vary versus the bridge output voltage. The total noise power, being the mean-square sum of the two, is dominated by the external sources when the bridge is unbalanced and is limited by the thermal noise level when the bridge is close to balanced state. The transition point between the two dominant noise regimes is denoted as

5

v b,o,t in Fig. 5 and is closer to the balanced state for an external source with higher IPN. In practice, the total noise is in many cases dominated by the external sources, since the phase noise levels of buffers and generators are much higher than the thermal noise level of the bridge, even for small bridge output voltages. As an example, consider a realistic case of the RF bridge as in Fig. 4, with G = 1 mS, C = 100 fF, G L = 0.01 mS, and C L = 1 fF (1% imbalance), driven at 1 GHz (ω = 2π · 1 G · rad/s) with an amplitude of v in = 1 V and read out at an observation time of 1 ms (ω = 2π · 1 k · rad/s). According to (3) and (10), the signal output of the bridge is v b,o = 2.5 mV, and the thermal noise power at the output 2 = 1.489 · 10−15 V2 . For an external source (driver is v th,n,o or generator) to contribute the same level of noise at the bridge output, a required IPN of −85.2 dBc is calculated from (11) or (12), which corresponds roughly to an SSB phase noise of −118 dBc/Hz over all frequency offsets below 1 kHz. This performance is at the boundary of what is achievable by the state-of-the-art frequency synthesizers at this frequency of operation [34], [35]. 3) Double-Balanced, Fully Differential Bridge: The single RF impedance bridge of Fig. 4, analyzed till this point, suffers from a large common-mode signal at its output. In order to achieve the highest sensitivity to load changes, (3) suggests that the drive amplitude voltage |v in | should be maximized. In a CMOS implementation, where the bridge is actively driven by MOS transistors, this maximum amplitude is in the order of the nominal supply (VDD). Moreover, the highest sensitivity is achieved when all branch nominal admittances are equal (Y0 ). Under these assumptions, the worst case common-mode signal v b,CM at the differential output of the bridge is half the supply voltage (peak-to-peak), on top of a useful differential signal v b,o , orders of magnitude smaller, as illustrated in the single-driven topology of Fig. 6(a). Such a large common-mode voltage poses a stringent requirement to the common-mode rejection ratio (CMRR) of the readout chain and compromises the linearity of the active circuitry following the bridge. An antiphase drive of each branch of the bridge, as shown in Fig. 6(b), can mitigate this problem, since the baseline signals, having a phase difference of 180°, will cancel out when combined at the output of the bridge, preferably capacitively to additionally achieve dc blocking. However, this results in a single-ended output of the bridge, and the benefits of a fully differential readout chain cannot be employed. Moreover, if the two drive signals are not exactly 180° out-of-phase, a phase mismatch signal (v b,M M ) will appear in the output of the bridge. This cannot be treated as a constant offset when this phase mismatch is load-dependent due to the limited driving capability of the bridge driver. A double-balanced configuration, shown in Fig. 6(c), uses an antiphase driven copy of the bridge (without the load connection). Capacitively combining the four bridge nodes (A to A and B to B , respectively) results in a differential output. Additionally, any signal caused by phase mismatch of the bridge drive turns into a common-mode signal, which is much smaller than VDD/2 and can easily be rejected in a

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Fig. 6. Evolution of single-driven, single-ended impedance bridge toward a fully differential double-balanced topology. v b,CM denotes for common-mode signal, while v b,M M is the signal caused by the phase mismatch between the two out-of-phase driving sinusoids. (a) Single. (b) Differential single-balanced. (c) Differential double-balanced.

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odd harmonics of LO, as shown in Fig. 7. Situated 2 f IF apart, these harmonics can be isolated and analyzed, enabling the characterization of the load at higher frequencies than the highest achieved by the fundamental drive, and at more than one frequency point at the same time. Since the amplitude of the higher order odd harmonics in the square wave reduces by at least 1/n compared with the fundamental order, where n is the harmonic, a lower sensitivity is expected at these higher harmonics. Nevertheless, useful information can still be acquired, contributing to the previously mentioned goal of redundancy. In addition to the baseband products of the mixing process, cross mixing can create content close to the even harmonics of f RF (e.g., 3 f RF − f LO ). Careful design of the mixer and LO signal is required to minimize self-mixing with the odd harmonics of LO, which will fall within the useful signal frequency f IF . In general, a fully differential chain with layout matching techniques can minimize the second-order harmonic content and nonlinearities. III. C IRCUIT D ESIGN In this section, we discuss the specific implementation and integrated circuit design of the permittivity sensor based on the previously reported architecture. The three circuit blocks comprising the sensor is the RF bridge, the downconversion mixer, and the bridge and LO drivers that provide the square wave for multiharmonic operation. A. Double-Balanced Fully Differential Bridge Design

Fig. 7. Block-level diagram of the multiharmonic IF downconversion architecture with annotated signals and their frequency-domain representation (inset).

fully differential chain. Nevertheless, using a double-balanced bridge configuration instead of a single one comes at the price of doubling both the area and the noise power as well as an increased power consumption needed for driving the bridge because the loading of the drivers is increased. C. Multiharmonic Downconversion The RF output of the bridge needs to be downconverted from the characterization frequency f RF to a convenient IF in order to be digitized and further analyzed. To achieve this, the bridge is connected to a downconversion mixer, as shown in Fig. 7, in which the output signal of the bridge is mixed with an LO signal at fLO , generating an output signal v IF , which is an exact replica of v b,o at f IF  f RF , assuming a perfectly linear mixing operation. A switching mixer with square-wave LO drive is preferred, as it can achieve a higher conversion efficiency than a smallsignal equivalent [36], [37]. As a result, the LO signal also contains odd higher order harmonics of the fundamental f LO . At the same time, it is convenient to apply a square drive to the bridge, in order to maximize its drive amplitude (signals DRIVE+ and DRIVE− in Fig. 7). Therefore, the bridge is driven at multiple odd harmonics, which will be downconverted to odd harmonics of f IF , after being mixed with the

The implemented sensing element is a square 100 × 100 μm2 -patch implemented in the top ultrathick metal of the CMOS stack (M7), with a nitride opening for direct interfacing with an MUT or used for probing. The patch also utilizes patterned thick metal layer (M6) connected to ultrathick through a large via, respecting all stress-related DRC rules for probing. This structure is EM-simulated in order to generate the RFM model of Fig. 3(b), discussed in Section II-A, which is later used for calibration of the system. Fig. 8 shows the schematic of the implemented bridge in which the sensor patch is embedded, implementing the fully differential double balanced architecture discussed in Section II-B3 with some additions for reconfigurability and practical considerations that will be discussed further. As seen in Fig. 8, the main part of the branch admittance is a capacitor Cb . In order to accommodate wide capacitive load variations and experimentally investigate the behavior of the bridge at various imbalanced states, Cb is implemented as a parallel combination of eight switchable capacitors. Each of these comprises a capacitor C1 of roughly 100 f F, in series with a 10-μm/40-nm CMOS switch. The capacitor bank is controlled by a unitary weighted 8-b digital signal b. Due to the finite quality factor of the capacitor and the equivalent ON/ OFF resistance of the switch, we can model the switched capacitor as an equivalent conductance in parallel with a capacitance, with varying values versus frequency during the ON- and OFF-state. Fig. 9 shows the simulated ON / OFF parallel conductance and capacitance versus frequency

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7

Fig. 8. Schematic of the implemented fully differential double-balanced bridge and 3-D view of patch implementation on two top metals of the 40-nm CMOS technology.

Fig. 9. Equivalent parallel capacitance and conductance of switched capacitor at ON and OFF states.

for the switched capacitor (postlayout extraction). The simulated ON-capacitance and conductance vary versus frequency from 130 to 100 fF and from 0.01 to 0.2 mS/GHz, respectively, while the OFF-capacitance and conductance are from 30 to 12 fF and 0.01 to 0.06 mS/GHz, respectively. At each frequency, the total branch capacitance and admittance depend on the number of ON capacitors, determined by the value of b as: Yb = b × YON + (8 − b) × YOFF . A proper value of b can be used to bring the branch admittance to a value, such that the balanced state falls close to the range of loads measured. For example, for the permittivity range of simulations in Fig. 3(b), we expect a load variation of 60–300 fF and 0–0.8 ms/GHz for patch capacitance and conductance, respectively. A value of b between 0 and 3 can fall within this range. A 1.2-k discharge resistor Rd is placed between the bridge middle nodes ( A, A , B, B  ) and ground in order to ensure a dc discharge path that sets the dc bias condition for the proper operation of the nMOS switches. The value of the resistor is a tradeoff between size consideration and minimum voltage drop due to bridge loading. Similarly, the four 25-fF combining capacitors Cc are of the same order of magnitude as the input capacitance of the mixer, for optimum voltage division. As suggested by (3), the output of the bridge is proportional to the amplitude of the drive signal v in . Since this value

depends on the supply voltage, it is desirable to decouple the system output from the bridge drive amplitude. In addition, in order to gain information of both capacitance and conductance, we need to acquire both the real and imaginary parts of the bridge output. Therefore, an amplitude and phase measurement of the bridge output is required. For the phase measurement to be consistent, a reference phase also needs to be measured. This is required in order to determine the relative phase variation at the output of the bridge, caused only by the patch load variation. A relative amplitude and phase measurement can be achieved without the introduction of any additional active circuitry, by disconnecting the bridge from the patch and connecting it to a fixed on-chip capacitance C f ≈ 100 fF, during a continuous-time measurement, through a series nMOS switch, as shown in Fig. 8. This switch operates in its linear region, because the discharge resistor Rd sets its dc bias point to zero, and the maximum voltage swing across the switch (350 mV in the presence of resistor Rd and parasitics to ground) is well below the simulated 1-dB compression point of 760 mV below 5 GHz. A digital control signal denoted as lc in Fig. 8 that controls the connection of the bridge to either the patch sensor the fixed capacitor C f , is used to acquire a continuous measurement trace containing the both outputs of the bridge during these two load-connection cases. The acquired signal is downconverted and digitized, and the two separate outputs are isolated in the digital domain by synchronization to the control signal lc. The fast Fourier transform (FFT) of the two outputs is then calculated and divided in order to acquire a consistent relative phase difference and amplitude ratio, which only depend on the relation between the fixed and the measured load. Note, however, that this solution does not eliminate the short-term variations of the bridge drive voltage that happen independently during the measurement of the two load-connection cases, as these variations are uncorrelated to each other. B. Downconversion Mixer Fig. 10 shows the schematic of the downconversion mixer connected to the bridge to perform a frequency translation of

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Fig. 10. Current-mode downconversion mixer schematic consisting of a transconductance stage and a current-mode switch quad.

the RF bridge output to IF. The topology implements a currentmode switching mixer that achieves low 1/ f noise operation and high linearity [38]. The transistors Q 1 and Q 2 , along with resistors R L , form a differential transconductance (gm ) stage. If the value of R L is large enough, most of the drain current of the transistors will be transferred to the output, converting the bridge output voltage (v RF+ , v RF− ) to a differential current (i RF+ , i RF− ). The transistor Q s sets the bias current, which is generally limited for two main reasons: 1) the large resistor value limits headroom of Q 1 and Q 2 , which is required for good linearity and 2) Q s needs to be small in order to minimize its parasitic drain capacitance that deteriorates the CMRR and the second-order nonlinearity. On the contrary, a higher bias current results in a larger amplification and, hence, a better noise performance. As a tradeoff, a bias current of 700 μA was chosen to achieve a transistor gm of 5 mA/V. The output current of the gm stage is fed to a CMOS switching quad that performs the mixing action. Capacitive coupling is used to prevent dc current through the CMOS switches, which is a source of flicker noise and nonlinearity [38]. An optimum switch size can be found, since a large size reduces the ON-switch resistance (and thus the insertion loss) but increases the parasitic capacitance to ground and the loading to the LO driver. In order to convert the downconverted signal current back to voltage and perform digitization of the waveforms using an A/D converter (ADC), a low-noise external transimpedance amplifier is used, which converts the current to voltage through a 10-k resistance and amplifies this voltage with a variable 0–40-dB gain. Fig. 11 shows the simulated conversion gain and noise figure of the mixer when terminated with an external 10-k resistance and driven by an input port with impedance equal to that of the bridge. To investigate the multiharmonic operation, the gain and the noise of the LO third- and fifth-harmonic components are also simulated. Due to the 1/n reduction in the LO amplitude, the conversion gain of the third and fifth harmonics is expected to be 9.5 and 14 dB lower than the first harmonic, respectively. This trend is seen at frequencies above 1 GHz while below that the first and third harmonics experience a larger loss in the RF path due to the capacitive coupling at the bridge-mixer and gm -quad connections. A 20-dB/decade gain roll-off is observed above 1 GHz. The noise figure is 7.5 dB at 2 GHz and stays below 10 dB in the GHz range. Below that, it increases

Fig. 11. Simulated conversion gain (top) and noise figure (bottom) versus of downconversion mixer with an IF of 150 kHz for three LO harmonics.

Fig. 12.

Schematic of the bridge and mixer drivers.

rapidly to 22 dB because of the signal loss at the bridge output capacitor Cc . As expected, the noise figure of the thirdand fifth-harmonic downconversion processes deteriorates by at least as much as the conversion gain deterioration. C. Square-Wave Drivers The bridge and LO drivers share the same topology that utilizes inverter amplifiers to achieve a square-wave rail-torail output. Shown in Fig. 12, the driver consists of a selfbiased inverter that sets the dc voltage of the input waveform to the desired mid-rail value by proper choice of the nMOS and pMOS size. Two complementary copies of the input are created, and a series of increasingly larger cross-coupled inverters further amplify the signal and ensure rise–fall edge alignment, thus minimizing phase imbalance. Optimization of the inverters’ transistor size ratio allows to minimize rise– fall mismatch that creates a common-mode voltage at the output of the bridge. In general, steeper edges (i.e., larger sized transistors and higher power consumption) minimize rise–fall mismatch across PVT variations. In fact, the simulated typical common-mode output on the bridge, caused by the driver at a 1.1-V supply, is 5 mV, while the worst case (fast-n/slow-p, VDD = 1 V) was simulated to be 20 mV, which poses no risk for the linearity of the gm stage, as would be the case with a large common mode caused by the use of a single bridge. Finally, the simulated IPN of the bridge driver, which contributes to the bridge output noise, is between −92 dBc at

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1 GHz and −81 dBc at 5 GHz, for an integration bandwidth of 0.01–1 kHz. IV. S YSTEM C ALIBRATION , ACCURACY, AND R ESOLUTION A. Calibration As discussed in Section II-B1 and summarized in (6) and (7), the real and imaginary parts of the inverse bridge differential output are the linear combinations of the weighted load conductance and susceptance. This result allows us to perform a linear fitting procedure for calibration. The benefit of such an approach is that it theoretically requires a minimum number of two known loads, if no systematic or random errors are induced by the calibration materials or measurement noise, although practically more points help average out such errors. In any case, a linear output expression alleviates any error induced from approximating (3) to a Taylor polynomial expansion of a certain order, bounded by the available number of calibration materials. Equations (6) and (7) hold true with the assumption that the bridge is perfectly balanced to the baseline load admittance, i.e., in the middle of the measured load range. In practice, however, due to the asymmetric nature of the patch node ( A in Fig. 8) to the rest of the bridge, and the finite quality factor of the switched branch capacitors, it is quite impractical to accurately ensure such a condition. A generic approach would be to assume that Y A = Yb + YOFF + Y L , where YOFF = G OFF + j ωC OFF indicates how much load should be added at the patch node, so that the bridge is balanced to the baseline load admittance. Being a fictional admittance, YOFF can assume both positive and negative values. The unbalance of the bridge can be defined as Y = YOFF + Y L . Assuming a linear behavior of the circuitry following the bridge, we can use the result of (6) and (7) and formulate the calibration equations about the measured chip output quantity out:   1 = K R + K G R · G w + K C R · Cw (13)  out and   1  = K I + K G I · G w + K C I · Cw (14) out where G w = G/|Y |2 , Cw = C/|Y |2 the unbalance weighted loads, and K R , K G R , K C R , K I , K G I , and K C I are real-valued numbers, further referred to as the K coefficients. A calibration operation would consist of the estimation of these coefficients as well as YOFF . Provided they are available, the sensor load Y L can be estimated by observing the respective chip output outm . More specifically, by solving the system of (13) and (14), the measured weighted load values are acquired Kˆ C I ({1/outm } − Kˆ R ) − Kˆ C R ({1/outm }− Kˆ I ) Kˆ C I Kˆ G R − Kˆ C R Kˆ G I (15) ˆ ˆ ˆ ˆ K G I ({1/outm } − K R ) − K G R ({1/outm }− K I ) = . Kˆ C I Kˆ G R − Kˆ C R Kˆ G I (16)

G w,m =

G w,m

9

From the definition of the weighted loads, we get G m =

G w,m 2 + ω2 Cw,m

(17)

Cw,m 2 G 2w,m + ω2 Cw,m

(18)

G 2w,m

and Cm =

from which, the measured load is calculated as Yˆ L ,m = Ym − YˆOFF . Although the approximate values of YOFF and the K coefficients can be estimated during the design process, their exact value remains unknown due to fabrication tolerances and modeling or simulation inaccuracies. In order to determine these values, a calibration procedure can be defined as follows. 1) Measure the sensor output at a set of known load values YL,cal . 2) Search for the combination of K coefficients and YOFF that achieve the best linear fit of C L ,m and G L ,m versus inverse output, according to (13) and (14), using the adjusted R 2 as a goodness-of-fit merit figure. 3) Store the combination of YOFF and K coefficients corresponding as the calibration parameters of the chip. Note that the calibration coefficients are frequency-specific, since both Y and YOFF are frequency-dependent (see Fig. 9). Moreover, even with the presence of mismatch of the branch admittances of the bridge, the calibration procedure still holds, because there always exists a YOFF , such that linear equations (13) and (14) still hold true. Therefore, minimizing mismatch during the design procedure is not a strict requirement, if YOFF is found through a search algorithm. B. Accuracy and Resolution A distinction should be made at this point between the accuracy and the resolution of the sensor. The accuracy of the permittivity measurement indicates its difference to the actual permittivity of the MUT, and it is affected by the temperature variation, the accuracy of reference liquids, and the accuracy of the assigned -to-Y transfer characteristic. Absolute accuracy is crucial for instrumentation applications, such as material characterization. In this paper, we make use of the tabulated permittivity values that originate from Debye models and that are accurate within 1% [39], [40]. Combined with the fact that no precise temperature is measured or imposed upon the MUT, the accuracy of the calibration procedure is expected to be at best within the same order. For the intended application of imaging, which requires contrast detection, we are rather interested in the measurement resolution, which relates to the minimum detectable permittivity variation and is directly linked to the noise levels at the output. Since the readout of the real and imaginary parts of the bridge output is done by measuring amplitude and phase, we need to link the resolution of the amplitude and phase readout to the noise level, and, from that, assess the expected system resolution. Let v IF = |v IF | · e− j φIF be the single-ended, amplified, and digitized voltage output of the chip. Assuming that the

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A/D conversion quantization noise is far below the signal noise, we can relate the minimum variance bound of the amplitude and phase, acquired by the FFT of v IF , to its signalto-noise ratio (SNR), through the Cramér–Rao bound [41] 2 var{|vˆIF |} ≥ v n,IF 2π var{φˆ IF } ≥ SNRIF

(19) (20)

2 where v n,IF is the noise at the system output. Using the 2 /(F · v 2 ), and the condefinitions for SNR, SNRIF = v b,o n,bo version gain CG = v IF /v b,o , and using, as explained in 2 2 + v2 = IPN · v b,o Section II-B2, that v n,bo th,n,o , we acquire

 2 2 var |vˆIF | ≥ IPN · F · v IF + CG · F · v th,n,o

 2 v th,n,o φˆ IF var ≈ IPN · F ≥ IPN · F + 2 2π v b,o

Fig. 13. Simulated   and   resolution versus permittivity for the chip model for f = 1 GHz, b = 2,  f = 1 kHz, I P N = −90 dBc, NF = 7.5 dB, and CG = 30 dB (off-chip amplification included).

(21) (22)

where F = 10NF/10 the system noise factor. Since a ratiometric measurement is carried out by dividing two chip output voltages (the output due to the measured load and the fixed capacitor output), we can infer, by propagation of uncertainty calculations, that the variance of the measured ratio signal out is 2 ˆ ≥ 2 · IPN · F · out2 + 2 · CG · F · v th,n,o var{|out|} 

φˆ out ≥ 2 · IPN · F. var 2π

(23) (24)

As expected, a larger external IPN and a system noise factor (F) incur a more noisy readout of both amplitude and phase. Moreover, an unbalanced bridge negatively affects only the variance of the signal amplitude while the phase stays, to a first-order approximation, unaffected, and only depends on the input noise and the noise performance of the readout circuitry. The variance of the measured amplitude and phase propagates to the real and imaginary parts and, through (1), (6), and (7), to a variance of the load (G and C) and permittivity, respectively. We can thus come to the conclusion that the optimal permittivity resolution of both the real and imaginary parts occurs when the bridge is perfectly balanced to the measured admittance. Indeed, as derived in (3) and (4), a balanced bridge has the highest Y L -to-output sensitivity (equal to 4Y0 /v in ). Moreover, the measured output variance is also minimized to the thermal noise level at balance, as predicted from (23). In order to evaluate the achievable permittivity resolution and confirm the optimum operating conditions, a perturbation analysis is carried out on the equations that govern the designed system [(3) multiplied by the system gain] using parameter values provided by the circuit-level simulations. A complex permittivity sweep is performed, and the calculated output amplitude and phase of the chip are superimposed by the random noise predicted by (23) and (24), respectively. Then, the calibration procedure is performed to evaluate the standard deviation of the permittivity and, hence, the resolution. The result of this procedure is surfaces such as the

Fig. 14. Simulated   (top) and   (bottom) resolution versus real part of permittivity, at different bridge capacitance settings for f = 1 GHz,  f = 1 kHz, IPN = −90 dBc, NF = 7.5 dB, and CG = 30 dB (off-chip amplification included).

ones in Fig. 13 for the simulated resolution of the real and imaginary parts of permittivity. For the specific bridge capacitance setting of b = 1 (approx. 260-fF branch capacitance and 42-μS conductance) and frequency of f = 1 GHz, there is a certain complex permittivity value that balances the bridge best, thus offering best resolution. As such, the complex permittivity resolution contains local minima at   ≈ 20.5 and   ≈ 19.5. Fig. 14 shows the simulated permittivity resolution versus MUT permittivity for various values of the branch capacitance setting b at 1 GHz. At this frequency, the best permittivity resolution is expected, since the noise figure and the external IPN of the used RF generator (Keysight E8257D) are minimum. By choosing the proper control value b, an absolute permittivity resolution of 97%) at 1.608 GHz would be delivered to the RF-dc converter which generates a dc voltage of 1.5 V. And then the low dropout voltage regulator is adopted to provide a supply voltage of 1.2 V for the rest of the building blocks. Biomedical signals would be first digitalized by the ADC and then the output would be transformed from 11-b parallel output to a single serial output. The DCU would deliver proper timing signals to control the operation of the system. Based on the injection-locked technique, a low-phasenoise carrier at 402 MHz can be generated by an injectionlocked frequency divider (ILFD) for data transmission by using a small amount of the 1.608-GHz RF input as the injection signal. Finally, the carrier and data are combined through OOK modulation scheme which is performed by using the data to modulate the ON/ OFF operation of the PA. III. C IRCUIT D ESIGN A. RF-DC Converter A RF-dc converter is used to convert the RF energy of the 1.608-GHz signal into the required dc power of the whole chip. As shown in Fig. 2, the RF-dc converter is constructed from three rectifiers in cascade, where each rectifier is based on a cross-coupled differential configuration in a bridge structure [14]. Consider the first rectifier: the dc component of the voltage waveform V X and VY in steady-state condition which can be regarded as the common-mode voltage is generated by the rectification operation and equals half of the output dc voltage of this stage. Particularly, this common-mode voltage act as a static gate bias that compensates the V th to improve the conversion efficiency, as mentioned in [14]. In addition, in this differential V th cancellation scheme, the gate voltage of each transistor is a differential-mode RF signal. When V X is negative and VY is positive, the nMOS M1 would behave like a forward-biased diode with a small ON -resistance. The turn-on voltage of this forward-biased diode would be effectively decreases because the gate bias voltage VY is positive. Meanwhile, the pMOS M4 behaves like another diode with a reduced turn-on voltage, so that the output dc voltage of this stage would be close to VY . As mentioned in Section II, the RF-dc would deliver a voltage of 1.5 V to the regulator which would generate a stable output voltage of 1.2 V. The RF-dc converter needs to provide

Fig. 3. Output power delivered by RF-dc converters containing 2–4 rectifier stages at different loads.

Fig. 4.

Schematic of the LDO.

an output voltage that is high enough to activate the regulator while maintaining good power conversion efficiency (PCE), which strongly depends on the number of the rectifiers in cascade. With a fixed input power of 6 dBm, the relationship between the output power and effective load is investigated for RF-dc converters containing 2–4 rectifiers, as shown in Fig. 3. According to the simulation results, the RF-dc converter formed from three rectifiers can deliver more power than those formed from two or four rectifiers. The required output power for the RF-dc converter to maintain the voltages of 1.2 and 1.5 V under different loads are also shown in Fig. 3. Notably, the RF-dc converter formed from three rectifiers can provide enough output power for the rest of the building blocks which drain the current of 0.97 mA from the supply voltage of 1.5 V and exhibit an effective load of 1.5 k. B. Low Dropout Voltage Regulator A low dropout voltage regulator (LDO) is used to convert the output voltage of the RF-dc converter into a stable supply voltage of 1.2 V required for the rest building blocks of the interface IC. The schematic of the LDO is shown in Fig. 4. The output voltage of the LDO can be expressed as   R1 + R2 Vin (1) + VREF Vout ≈ R2 AEA where AEA represents the gain of the error amplifier. Apparently, the output voltage Vout is independent of the input voltage Vin when AEA is large enough. A reference voltage VREF of 0.6 V is generated by a supply voltage independent bias circuit so that an output voltage of 1.2 V can be obtained when the resistors R1-2 (101 k) are identical.

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Fig. 5.

Fig. 6.

3

Schematic of the ILFD.

Fig. 7.

Block diagrams of the SAR-ADC and the DCU.

Fig. 8.

Measured input returned loss of the RF-dc converter.

Schematic of the PA.

C. Injection-Locked Frequency Divider The injection-locked divide-by-four circuit is formed from two injection-locked divide-by-two circuits in cascade. The schematic of the injection-locked divide-by-two circuit is shown in Fig. 5 [15]. The divider is transformed from a threestage ring oscillator. The bias current, as well as the selfoscillation frequency of the ring oscillator, is determined by the bias voltage (Vbias ). As shown in Fig. 5, the voltage at node x (Vx ) would be close to the voltage at node y (Vy ) at the instants of zero-crossing during each oscillation period of the free-running ring oscillator. The injection signal is applied at the gate of the nMOS M7 which functions as a switch with a bias voltage Vinj_bias to reduce the ON-resistance. When the circuit operates in states A and C, the input voltage Vinj reaches its maximum so that the ON-resistance of M7 comes to its minimum and the voltages at nodes x and y would be equalized. In state B, the input voltage Vinj reaches its minimum so that M7 turns off and the ring oscillator oscillates at the self-oscillation frequency [14]. Through the operation of M7 , the instants of zero-crossing can be controlled by the injection signal. If the self-oscillation frequency of the ring oscillator is close to half of the injecting frequency, the circuit would most likely be injection-locked and function as a divideby-two circuit to deliver an output at half of the injecting frequency.

M1 and M2 , respectively. Through the load–pull technique, the optimal load impedance of the PA is found. According to the simulation results, the 50  is converted into this optimal load impedance (15 + j140)  with an off-chip matching network. E. Successive Approximation Register ADC and Digital Control Unit A single-ended-input SAR-ADC is used to convert the biomedical signal into its digital form. As shown in Fig. 7, it consists of a sample-and-hold circuit(S/H), a capacitive digital-to-analog converter (Cap-DAC), a voltage comparator, and a successive approximation register (SAR). It operates at a sampling rate of 50 k-samples/s. The DCU consists of a PISO circuit, a system timing control circuit and multiplexers, as shown in Fig. 7. The PISO circuit is used to deliver the 11-b output of the SAR-ADC through a serial port. To enable the OOK modulation, two voltages 0.612 and 0.542 V from the bias circuit are fed to the multiplexers whose outputs VBP /VBN would be either 0.612/0.542, or 1.2/0 V, depending on the serial output from the SAR-ADC, DATA.

D. PA For an implantable system, the efficiency of the PA should be optimized for low-power operation. The class E PA can achieve the highest efficiency among others, but the large transistor size would cause a heavy load to its preceding stage. Therefore, the class AB PA is adopted in this system. The schematic of the class AB PA is shown in Fig. 6. The PA would be turned on to deliver logic 1, at which the bias voltages Vbp of 612 mV and Vbn of 542 mV are applied to

IV. E XPERIMENTAL R ESULTS A. RF-DC Converter An impedance matching -network is off-chip deployed at the input of the RF-dc converter. The input returned loss from 1 to 2 GHz is shown in Fig. 8, where the input returned loss falls below −10 dB from 1.53–1.67 GHz. The PCE versus load resistance at different RF powering levels is shown in Fig. 9. The efficiency grows from 17.4%

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

Fig. 9.

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Measured PCE of the RF-dc converter at different loads.

to 25.7% as the input powering level is increased from 6 to 10 dBm. At input powering level of 8 dBm, the circuit achieves the efficiency of 25.7% for the effective load resistance of 1.5 k. The efficiency is higher than 25% as the powering level exceeds 8 dBm. Since the power consumption of the whole chip is around 1.44 mW, an external RF signal with the power level of 8 dBm can provide enough energy for the rest building blocks of the interface IC. B. Low Dropout Voltage Regulator During the measurement, the supply voltage of the LDO is provided by a power supply and varied from 0.9 to 2 V, while the rest building blocks totally consume the current around 1 mA. The output voltage would remain 1.2 V when the supply voltage exceeds 1.2 V, as shown in Fig. 10(a). The dynamic load regulation of the LDO can be observed from the variation (Vout ) in the transient output voltage, as shown in Fig. 10(b) and (c). The output stream from the PISO is sent to the data input of the OOK transmitter for modulation. When a logic 1 is delivered to the data input, the transmitter drains more current from the LDO, which causes a heavyload condition. When a logic 0 is delivered to the data input, the transmitter drains less current from the LDO, which causes a light-load condition. As shown in Fig. 10(b), the LDO output voltage exhibits small variations (1-mW Peak Output Power Employing Common-Mode Impedance Enhancement Faisal Ahmed , Student Member, IEEE, Muhammad Furqan, Student Member, IEEE, Bernd Heinemann, and Andreas Stelzer, Member, IEEE

Abstract— We present a novel method of maximizing the output power and efficiency of millimeter-wave and terahertz signal sources, which are based on the push–push topology. In this method, the common-mode impedance of a differential Colpitts oscillator operating in the odd mode is maximized by introducing a fixed-valued capacitor (C r ) at the common-base node. This capacitor is designed to introduce a common-mode parallel resonance at the desired second harmonic, boosting the common-mode voltage swing and subsequently its output power. The proposed method is analyzed using a high-frequency evenmode π-model. Analytical expressions of input impedance are derived and are used for calculating the common-mode resonance frequency and the required value of C r . Two 0.3-THz voltagecontrolled oscillators (VCOs) are implemented in a 130-nm SiGe BiCMOS process. It is shown that by using the proposed technique, the output power is improved by more than 6 dB, as compared with the conventional approaches. The implemented VCOs work from 292 to 318 GHz and 305 to 327 GHz, delivering a peak output power of 0.6 and 0.2 dBm, with a dc-to-RF efficiency of 0.8% and 0.9%, and can achieve a phase noise of −108 and −105 dBc/Hz at 10-MHz offset, respectively. As compared with the prior state-of-the-art Si-based tunable signal sources and arrays working above 270 GHz, this paper shows the lowest phase noise and the best figure-of-merit, while having an excellent output power, a tuning range, and a dc-to-RF efficiency. Index Terms— Heterojunction bipolar transistor (HBT), millimeter wave (mm-wave), SiGe BiCMOS, terahertz (THz), voltage-controlled oscillators (VCOs), wide tuning range.

I. I NTRODUCTION

T

ERAHERTZ (THz) integrated circuits and systems based on III-V semiconductors, such as InP [1], InAs [2], InGaAs [3], and advanced Si/SiGe-based technologies [4] have shown an enormous potential in terms of feasibility,

Manuscript received May 24, 2017; revised August 9, 2017; accepted October 7, 2017. Date of publication November 8, 2017; date of current version March 5, 2018. This work was supported by the Austrian Center of Competence in Mechatronics. (Corresponding author: Faisal Ahmed.) F. Ahmed and M. Furqan were with the Institute for Communications Engineering and RF-Systems, Johannes Kepler University, 4040 Linz, Austria. They are now with Danube Integrated Circuit Engineering GmbH, 4040 Linz, Austria (e-mail: [email protected]). B. Heinemann is with Innovations for High Performance Microelectronics, GmbH, 15236 Frankfurt (Oder), Germany. A. Stelzer is with the Institute for Communications Engineering and RF-Systems, Johannes Kepler University, 4040 Linz, Austria. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2767593

performance, and lower cost compared with photonics-based alternatives [5]. The potentially immense and unregulated bandwidth available from 0.27 to 3 THz presents a major motivation for several Gb/s to 100-Gb/s wireless communication systems. Wireless channels have minimum latency and are more suitable for real-time systems. THz wireless systems have been envisaged for terrestrial high-capacity Tb/s links over distances of greater than 1 km as well as for indoor WLAN with Gb/s speed [6]. Practical communications over a distance of 20 m at a data rate of 100 Gb/s have been demonstrated using a combination of THz photonics and 35-nm metamorphic high electron mobility transistor technology with a power-gain cutoff frequency ( f max ) of 900 GHz [7]. Researchers have recently demonstrated digital data transmission over the 300-GHz band at a rate exceeding 100 Gb/s over a single channel using a 40-nm CMOS process [8]. In terms of space applications, theoretical models have shown that even with severe atmospheric attenuation, geostationary THz satellite links can support data rates up to 1 Tb/s [9]. In addition to vast potential in the field of wireless communications, the lower-THz regime finds inherent advantages in many other diverse applications, such as noninvasive imaging radars for standoff personal security/screening, material and molecular identification, medical diagnosis, and remote sensing [10], [11]. These applications are possible because of the unique characteristics of THz waves, such as their nonionizing nature, penetration ability, and specific frequency-dependent absorption and dispersion properties [12], [13]. Furthermore, in imaging applications, the available wide bandwidths in this frequency range has the potential to reach even submillimeter range-resolution [14], [15]. In order to successfully attain any of the aforementioned applications, one of the key challenges includes the development of high-power-integrated THz sources working within the atmospheric transmission windows [6]. These signal sources are indispensable for transmitters and receivers. Fundamental signal sources based on common-base crosscoupled topology working above 300 GHz with more than 4.3-dBm output power have been demonstrated using a 250-nm InP-based process with a current-gain cutoff frequency ( f T ) and f max of 392 and 859 GHz, respectively [16]. Using the same technology, two differential oscillators

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AHMED et al.: 0.3-THz SiGe-BASED HIGH-EFFICIENCY PUSH–PUSH VCOs WITH >1-mW PEAK OUTPUT POWER

working at 280 GHz were combined using rat-race and the Wilkinson power combiner to achieve an output power of 10 dBm [17]. An experimental SiGe heterojunction bipolar transistor (HBT) technology with f T / f max of 550/720 GHz was recently presented [4]. This shows that in the near future, Si-based alternatives will enable fundamental-mode circuits and systems to work even above 0.3 THz. For now, harmonic extraction continues to be an indispensable technique to obtain sufficient output power at THz. At these frequencies, the available gain from the transistors is very low and passive devices exhibit much higher losses. These fundamental limitations have led to the development of new techniques and architectures for designing high-power THz frequency sources. In [18], a maximum-gain ring oscillator topology was introduced maximizing the power gain by means of appropriately designed passive matching networks. This increase in gain helps to achieve higher oscillation frequency. However, the topology is limited to ring oscillators, which is suitable mostly for CMOS-based designs. To maximize the output power of the harmonics, traditional load–pull simulations were utilized, beyond which the power-combining network was used to sum the power from multiple stages resulting in more dc power consumption and complexity. Building up on this approach, a cross-coupled push–push voltage-controlled oscillator (VCO) working around 239 GHz with −4.8-dBm output power was reported in [19], which replaces the passive matching network with transformer-based resonators. Similarly, capacitive feedback frequency-enhancement, which utilizes both negative resistance Colpitts approach and a capacitive feedback similar to that of −G m oscillators, was reported in [20]. An approach for increasing the oscillation frequency beyond the device cutoff frequency of f T was published in [21], by using a frequency-selective negative resistance tank. A fundamental oscillator prototype based on the buffer-feedback topology, working at 300 GHz in a 65-nm CMOS technology, was demonstrated by Razavi [22]. Varactors used for frequency tuning in the millimeterwave (mm-wave) region exhibit very low quality factors, severely limiting the output power. Furthermore, the parasitic capacitances start dominating at these frequencies, decreasing the effective tuning range. To address this challenge, power combining and harmonic extraction from coupled oscillators with multiple cores have been proposed [23]–[26]. However, the maximum tuning range is still very limited and many cores have to be used to achieve enough output power. Recent publications have demonstrated CMOS and SiGe HBT-based oscillators/VCOs with power sufficient enough for useful system applications. These include push–push oscillators [19], [27], [29], fundamental-oscillator, doubler hybrids [27], [29], [30], and triple-push oscillators [31], [32]. High tuning range and output power in the THz are attainable using frequency multipliers [33]–[37]. However, high-power external signal sources are required to drive the frequency multipliers into saturation. In this paper, we present two push–push VCOs based on a novel technique of introducing a common-mode resonance in a Colpitts push–push oscillator circuit aimed to maximize

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the harmonic output power manifolds as compared with a conventional approach. The proposed technique does not influence the odd-mode impedance and thus the fundamental-mode operation of the circuit can be independently designed. Prior works based on CMOS have relied on maximizing harmonic power by increasing the voltage swing at the fundamental frequency using a self-feeding line and blocking the desired even-harmonic signal leakage to the gate [23]–[25], [38]. Other works have utilized the conventional load–pull simulations to find the optimum condition for maximum even-harmonic power generation [18], [19], [29]. The VCOs presented in this paper operate from 292 to 318 GHz and 305 to 327 GHz and provide a peak output power of 1.15 and 1.05 mW, respectively (including pad losses), which to the best of our knowledge represent the highest output power reported for any Si-based tunable signal source based on a single oscillator operating above 270 GHz. Both VCOs demonstrate state-ofthe-art dc–RF efficiencies of 0.8% and 0.94%, respectively. The technique proposed in this paper can be employed to all oscillator and cascode-based frequency multiplication circuits relying on push–push-based even-harmonic extraction. This paper is organized as follows. Section II provides an analysis of the proposed technique using a high-frequency even-mode equivalent circuit of a Colpitts differential oscillator. Using both analytical expressions and Spectre simulations of the common-mode input impedance, the efficacy of the technique in improving the output power is explained. Technology details, with emphasis on high-speed HBT devices and varactor characterization, are provided in Section III. In Section IV, we focus on the details of the THz VCO circuit design, parasitics, and layout issues, with emphasis on the common-mode impedance design methodology. In this section, we also discuss the influence of the proposed technique on tuning bandwidth, frequency pulling, and phase-noise performance. The detailed characterizations and a thorough comparison with state-of-the-art sub-THz and THz signal sources are provided in Section V. Section VI contains the conclusion and summary. II. A NALYSIS OF THE P ROPOSED T ECHNIQUE A simplified schematic of a push–push VCO based on the differential Colpitts topology is presented in Fig. 1(a). L B is the tank inductance and r B comprises the inductor series resistance as well as the intrinsic base resistance. Cr is a fixedvalued resonant capacitor proposed in this paper for commonmode impedance enhancement. Conventionally, a high-valued bypass capacitor is used instead of Cr for providing a lowimpedance path. For the oscillator analysis, we have selected a high-speed HBT (Q 1 ) having five emitter fingers with 2-μm length each. The reason for selection of this transistor size will be provided in Sections III and IV. Following the procedure described in [39], the small-signal equivalent-circuit parameters for the calculated plots are directly extracted from the VBIC model of the HBT at 300 GHz. These parameters, along with the simulated values of f T , Ibias , and gm , are provided in Table I for reference. The HBTs are biased around their peak- f T current density. The values of L B and Cvar are designed to generate a fundamental odd-mode oscillation

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TABLE I D ESCRIPTION AND VALUES OF THE S MALL -S IGNAL E QUIVALENT-C IRCUIT PARAMETERS

The model is simplified by combining the impedances as shown in Fig. 1(c). We denote the impedance looking into  , the common node without and with Cr as Z in and Z in respectively. The input impedance Z in is given by Z in = {Z in } + j {Z in} = Z B + Z π  Z μ

(1)

Z π = Z π + (β(ω) + 1)Z E .

(2)

where The frequency-dependent small-signal gain β(ω) is given by β0 (3) β(ω) = 1 + j ωωT β0

Fig. 1. (a) Simplified lumped element circuit of a differential Colpitts oscillator-based push–push VCO. The blue dashed line shows the proposed common-mode resonant capacitor Cr . (b) High-frequency equivalent evenmode π -model of the VCO half-circuit shown in (a). (c) Simplified equivalent  (with C ). circuit for calculating the input impedance Z in (without Cr ) and Z in r (d) Equivalent input-impedance of the circuit depicted in (c), with and without Cr , showing Reff and Ceff .

at 150 GHz, which is half the desired output frequency of 300 GHz. The impedance seen at the common nodes of the oscillator should be properly designed to maximize the power of the second harmonic [40]. In order to analyze the input impedance of the Colpitts oscillator, we consider its high-frequency even-mode equivalent π-model, as shown in Fig. 1(b). For odd-mode operation, the voltage waveforms are added out of phase and a virtual ground is therefore presented at all of the common nodes. For even mode, the voltage waveforms add in phase, resulting in half the capacitance or twice the inductance/resistance at the common nodes [see Fig. 1(b)].

where β0 corresponds to the small-signal low-frequency gain, and ωT = 2π f T . For ω  ωT /β0 , which holds true in this paper, β(ω) can be approximated as ωT . (4) β(ω) = − j ω The parallel combination of Z π and Z μ is the effective impedance of the active device and the varactor and is denoted by Z eff , as shown in Fig.1(c). To determine the oscillation region and understand its behavior over frequency, it is useful to separate Z eff into its real and imaginary components 1 Z eff = Reff + j X eff = Reff + . (5) j ωCeff The expressions for calculating Z eff , Reff , and Ceff , using the values given in Table I, are given in (6)–(8), shown at the bottom of the next page. The input impedance of the oscillator circuit without Cr can therefore be represented as a simple  , and C , as series resonance circuit consisting of L B , Reff eff shown in Fig. 1(d). Fig. 2 shows the plots of Reff and Ceff as a function of frequency. We can clearly see that both Reff and Ceff are frequency-dependent. The condition for oscillation is met when the negative resitance Reff compensates the resonator losses at the oscillation frequency. At the startup, this negative resistance must exceed the losses by about 20% [41]. The fundamental resonant frequency is then determined by the values of L B and Ceff . Now, we can express the input impedance of the oscillator with and without Cr , using these effective impedances as follows: 1 Z in = Reff + r B + j ωL B + (9) j ωCeff

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Fig. 2. Calculated plots of effective resistance (Reff ) and effective capacitance (Ceff ) as a function of frequency.

and  Z in

 =

Reff

1 + r B + j ωL B + j ωCeff

 

1 . j ωCr

(10)

The input impedance graphs of the push–push VCO as a function of frequency (shown in Figs. 3 and 4) are plotted using the analytical expression of the input impedance and using Spectre simulations of the even-mode half-circuit of Fig. 1(a). We have made the comparison in order to validate the analysis and show that the simplified, high-frequency circuit model provides not only a good correlation to the Spectre simulations based on the complex VBIC model but also provides better insight. The behavior of the input reactance {Z in } without Cr is plotted in Fig. 3(a). It shows that the reactance of the circuit is zero around the resonance frequency of 150 GHz and starts to become inductive with increasing frequency. At the desired second harmonic frequency, the circuit shows a highly inductive reactance. In order to decrease this reactance, a parallel common-mode resonance f CM is introduced by connecting the capacitor Cr with a properly chosen value in parallel. This resonance is seen in Fig. 3(b) around 300 GHz. Since the capacitor Cr is connected to a common-node, it will ideally have no influence on the fundamental ( f osc = f o /2) mode operation of the circuit. The real part of input impedance {Z in } with and without Cr is plotted in Fig. 4. Without Cr , the plots show no resonance and the resistance of the circuit increases monotonically with the frequency and becomes positive above 200 GHz. Around the second harmonic frequency, the resistance is less than 8 . However, when Cr is employed, it introduces a strong resonance around the second harmonic and increases

Z eff = Reff = Ceff =

(Cπ Cvar (Cπ Cvar

Fig. 3. Analytical and simulated plots of the input reactance of the push–push VCO, as seen from the base terminal. (a) No Cr is used. (b) Optimized value of Cr is used for common-mode impedance enhancement.

the common-mode resistance by more than 15 times. This higher even-mode dynamic impedance helps to generate a stronger second-harmonic signal at the common nodes. The generated power can then be delivered to the output load and extracted from either the base, collector, or emitter, depending upon the optimum load-line impedance.  }, we can Considering Fig. 1(d) and maxima of {Z in calculate the common-mode resonance frequency fCM in terms  = r + R , L , C , and C of the lumped parameters, Reff B eff B eff r  2 1 2(Ceff + Cr )L B − Ceff Cr Reff f CM = . (11) 2π 2Ceff Cr L 2B For an mm-wave LC oscillator circuit design, the values of capacitance as compared with the values of inductance are usually smaller by a factor of thousand or more. Therefore, the

− j (Cπ + Cvar )rπ ω2 − (1 + Cπ rπ ωT )ω + j ωT + Cμ (Cπ + Cvar ))rπ ω3 − ( j Cμ Cπ rπ ωT + j (Cμ + Cvar ))ω2 − Cμ ωT ω Cvar (Cvar − Cπ2 rπ ωT )rπ ω2 − Cvar ωT + Cμ (Cπ + Cvar ))2rπ2 ω4 + ((Cμ + Cvar )2 − (Cμ2 rπ ωT (2Cvar − Cπ2 rπ ωT )))ω2 + Cμ2 ωT2

(Cπ Cvar + Cμ (Cπ + Cvar ))2rπ2 ω4 + ((Cμ + Cvar )2 − (Cμ2 rπ ωT (2Cvar − Cπ2 rπ ωT )))ω2 + Cμ2 ωT2 (Cπ + Cvar )(Cπ Cvar + Cμ (Cπ + Cvar ))rπ2 ω4 + (Cμ + Cvar − Cμ rπ ωT (2Cvar − Cπ2 rπ ωT ))ω2 + Cμ ωT2

(6) (7) (8)

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Fig. 5. (a) Layout of a high-speed HBT device with a CBEBC configuration showing the interdigitated structure. (b) Interconnect aluminum layers from Metal 1 (M1) to Thick Metal 2 (TM2), drawn to scale. Vias are not shown for simplicity.

Fig. 4. Analytical and simulated plots of the input resistance of the push–push VCO, as seen from the base terminal. (a) No Cr is used. (b) Optimized value of Cr is used for common-mode impedance enhancement.

second term in the numerator of (11) can be neglected and the equation can be simplified to  1 Ceff + Cr f CM ≈ . (12) 2π Ceff Cr L B With this equation, we can approximate f CM , using values of Cr and L B from Table I, and value of Ceff from Fig. 2. The calculated f CM is equal to 319.6 GHz, which is within an error of 6% as compared with the simulated value. Using either (11) or (12) and considering f CM to be equal to the desired second harmonic output frequency f o , we can calculate the required value of the proposed resonant capacitor Cr . III. SiGe HBT BiCMOS T ECHNOLOGY The VCOs based on the proposed technique are manufactured in a 130-nm SiGe HBT BiCMOS technology SG13S from Innovations for High Performance Microelectronics (IHP) [42]. It features bipolar devices based on SiGe:C npn-HBTs with an f T / f max of up to 250/300 GHz. Two different high-speed bipolar devices are available, one with a BEC layout configuration and a maximum emitter length of 0.84 μm, while the other device has a CBEBC configuration with a maximum emitter length of 2 μm. The device with the CBEBC configuration offers somewhat a higher f max value.

The HBTs have a dc current gain of around 900, a collector– emitter breakdown voltage (BVCEO ) of 1.7 V, and a collector– base breakdown voltage (BVCBO ) of 5 V. The maximum output voltage swing for safe operation is between BVCEO and BVCBO , depending upon the base current [43]. Both types of devices can have up to eight emitter fingers in parallel. This interdigited structure helps in establishing a uniform current density across the base region and also helps to reduce the series resistance. The process offers dual gate oxide thickness for 1.2- and 3.3-V supply, offering nMOS, pMOS, and isolated nMOS transistors. The backend-of-line consists of seven aluminum interconnect layers comprising five thin layers and two thick top metal layers of 2 and 3 μm. The layout and the metal stack for an HBT with a CBEBC configuration and a size of 0.17 × 2 × 5 μm2 are shown in Fig. 5. The contact and via losses up to the top thick metals decrease the effective available gain and fmax of the HBT, which have a direct influence to the center frequency and output power of the designed oscillator [44]. Fig. 6 shows the simulated maximum available gain (MAG) and f T / f max for two different HBT sizes with the CBEBC configuration. The devices are simulated around the peak- f T current density at VCE = 1.2 V. The plots include the via and wiring parasitics, extracted using the Sonnet EM solver. The MAG of both devices is less than 3 dB beyond 160 GHz, with the smaller device showing comparatively higher gain. However, the larger HBT shows better performance in terms of f max . We have used both these device sizes for designing the proposed VCOs with a similar target output frequency of 0.3 THz. The process offers accumulation-depletion mode differential MOS varactors with thick gate oxide (7 nm). An MOS varactor compared with a junction varactor exhibits higher capacitance ratio but is more prone to low-frequency noise [45]. A differential structure offers advantage in terms of a quality factor, a tuning range, and a Si area [46]. The process offers MOS varactors with an area of 3.74 μm × 0.3 μm, with multiple gate fingers. A cross-sectional view of an MOS varactor provided by the IHP process is shown in Fig. 7. The simulated and measured results for the MOS varactor with ten gate fingers are presented in Fig. 8. We can observe in the measured curves plotted in Fig. 8(a) that beyond the strong decrease of the

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Fig. 8. Simulated and measured characteristics of the MOS varactor. (a) Capacitance. (b) Q-factor.

Fig. 6. Simulated plots of (a) MAG of HBTs biased at peak- f T current density and (b) f T / f max . The HBTs are 0.17×2×4 μm3 and 0.17×2×5 μm3 in size. Plots include the effect of parasitics up to the TM2.

Fig. 7. Cross-sectional view of the accumulation-depletion mode differential MOS varactor.

capacitance, a slight reduction is still possible in the depletion region by increasing the control voltage Vctrl . The varactor shows a measured Q-factor of around 10 at 66 GHz. At higher control voltages, the quality factor is improved. IV. THz VCO S C IRCUIT D ESIGN A. Topology Selection There are various debates when it comes to selecting the best topology for the realization of an mm-wave push–push VCO. The conventional differential Colpitts

topology is usually preferred for its phase-noise performance and its ability to operate close to f max [50], [51]. A wider tuning range can be achieved by using an additional varactor at the base terminal of the HBT, a configuration known as the Colpitts–Clapp topology [29], [52]. However, both conventional and transformer-based cross-coupled topologies have been shown to achieve excellent performance in terms of efficiency and wide bandwidth in bipolar and CMOS technologies [19], [27], [53]. For oscillators based on the push–push topology, the second harmonic output can be obtained at any of the virtual differential ground nodes, which means that output can be extracted from either the base, collector, or emitter terminal [54]. This selection depends on several factors, such as the available voltage swing, the common-mode impedance seen at the node, the effect of load pulling, as well as the phase noise performance. Usually large signal simulations are used for finding the final optimum configuration. In [55] and [56], a push–push output is extracted from a common-node of the coupled emitter network. However, in most cases of the mm-wave VCO design, the common-node at the coupled emitter network does not provide the largest signal swing and/or the optimum impedance for maximum power transfer. In comparison, in the case of a common-collector (CC) topology, extracting the output from the base node has the advantage of having an efficient layout as it only requires a single inductor per oscillator; this is usually the most areaconsuming component. In addition, a CC topology provides excellent instability and usually good output power [57], [58]. The output extraction from the collector in a Colpitts configuration is largely advantageous, because the load is isolated from the resonant tank (present at the base and the emitter). This alleviates load pulling and unwanted external locking (frequency steps in the tuning curve) due to reflections from an unmatched load, as described in [51]. This topology also usually provides the largest harmonic power due, primarily, to the reason that a major portion of the second harmonic current from the core HBTs will flow to the collector nodes [59]. Similarly, for a cross-coupled topology, it has been demonstrated using analysis and experimental results that a common-base configuration (with output taken at the collector) outperforms the conventional common-emitter configuration in terms of achieving higher oscillation frequency and output power [16]. A fully differential Colpitts topology is selected for two designed VCOs, which are designated as VCO1 and VCO2

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where f m = f − f osc . Usually, for oscillators working above 100-GHz fundamental frequency, the MIM capacitor placed in parallel to the base–emitter junction of the transistor (Q 1 in Fig. 9) has to be removed in order to operate at higher frequencies. Furthermore, keeping Cπ  Cvar (either by selecting a larger HBT size or a smaller varactor) helps in reducing frequency pushing and phase noise and improves the tuning range. However, at mm-wave frequencies, larger HBTs lead to a decrease in dc-to-RF efficiency because of saturation of the available gain with increasing HBT size and if very small varactors are selected, their relative parasitics become too overwhelming. We therefore selected a size of the HBTs while considering the tradeoffs among the available transconductance, the oscillation frequency, and the phasenoise performance. The varactor size was selected by bearing in mind its quality factor, the tuning bandwidth, and the size of the HBTs used. B. Design of the Common-Mode Impedance and Load Matching

Fig. 9. Schematic of the push–push VCO. The contact parasitics denoted by L par and Rpar are also shown.

in this paper. Both VCOs target the same output frequency of around 0.3 THz. VCO1 is optimized using smaller HBT and varactor size as compared with VCO2. This allows us to observe the influence of HBT/varactor geometries on the output power, efficiency, tuning range, and the phase-noise performance. The HBTs are biased at their peak- f T current density. The circuit schematic including the bias and tuning network is provided in Fig. 9. Since the desired mode of oscillation is odd mode (ωosc = ωo /2), the simplified fundamental oscillation condition is given by [40] Gm = Rpar + r B + R S (ωo /2)2 Cπ Cvar

(13)

where Rpar , R S , and G m are the parasitic resistance of the contact via, the series resistance of TL B , and the large signal transconductance, respectively. Operating near f T , limited G m can be a challenge. In addition, the available gain and the effective f max because of the parasitic losses of the HBT contacts to the topmost RF thick layers are further decreased. The desired output frequency ωo is given by    1 1 Cπ Cvar = (L TLB + L par ) Cμ + . (14) ωo 2 Cπ + Cvar This equation suggests that increasing the size of the HBT would lead to a decrease in the oscillation frequency. The values of Cπ and Cvar , together with the oscillation amplitude and the total equivalent input noise current of the HBT (In ), determine the VCO phase noise performance, as given by [60], [61] L( fm ) =

Pnoise |I 2 | = 2n 2 Po Vosc fm

Cπ2



1 Cπ C var

+1

2

(15)

Although the push–push oscillator is a strongly nonlinear circuit that operates under large signal conditions, a very efficient way to design the common- and differential-mode impedance is to start with a small-signal S-parameter analysis. The simulated differential and common-mode impedance Z D and Z CM , looking into the base of the transistors (as annotated in Fig. 9), are plotted in Fig. 10. Since the design procedure is similar to both VCO1 and VCO2, simulation results from only VCO2 are presented here. The circuit is optimized to produce a large negative resistance at the fundamental frequency, which is essential to compensate all the tank and parasitic losses. A large negative resistance helps in generating a high oscillation voltage. To introduce the common-mode resonance f CM at f o , capacitor Cr is connected at the base and the initial value is calculated using (11). As can be seen, a large negative resistance is produced at 150 GHz, which would ensure that the condition for the onset of oscillation is satisfied. Z D will not be influenced by Cr , as it is connected to a differential virtual ground node. The behavior of Z CM is more interesting as it shows a distinct parallel resonance at 300 GHz. The Q-factor of the common-mode resonance estimated from the magnitude of Z CM impedance plot is around 4. On one hand, the higher voltage swing caused by the enhanced Q-factor in the common-mode impedance leads to increased harmonic power. On the other hand, the Q-factor is not high enough to considerably reduce the bandwidth of the output matching network. The optimum load-line impedance Z opt , which would fulfill the optimum power-matched condition Popt for the second harmonic, is found by load–pull simulations. The output signal is extracted from the common node at the collector. The collector is connected to the power supply via an RF choke. Fig. 11(a) shows the load–pull results obtained by using harmonic-balance simulations when no Cr is used to enhance the common-mode impedance. The power contours represent output power from −12 to −6 dBm with a step size of 1 dB. As a result, the maximum Popt that could be obtained in this case is around −5 dBm. The procedure is repeated with exactly the same circuit conditions but using a fixed value

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Fig. 10. Simulated input impedances as seen from the ports Z D and Z CM for the designed THz VCO. The fundamental resonance can be seen at 150 GHz with a negative resistance of around −160  and a commonmode resonance achieved using an optimized value of Cr is found at around 300 GHz. (a) Imaginary part. (b) Real part.

of Cr to maximize Z CM , as shown in Fig. 11(b). The power contours are plotted with a 1-dB step from −10 to 0 dBm, with a peak second-harmonic power of +1.7 dBm. This shows an improvement of 6.7 dB in output power with the same dc power consumption as compared with the case if the proposed common-mode impedance enhancement technique is not used. The 50- load is matched to Z opt using TLC , TLo , and CC . The transmission line TLstub is used for compensating the RF-pad capacitance. Fig. 12(a) shows the voltage swing of the second-harmonic signal at the load R L for the two optimized VCOs with and without Cr . The improvement in the pk–pk voltage swing is almost 2.5 times as compared with the case if no Cr is used. The voltage swings correspond to around 1.5- and −6-dBm ouput power at R L . Fig. 12(b) clearly demonstrates how the fundamental and the second-harmonic power vary with respect to Cr . As expected, the fundamental power remains almost constant, however, the second harmonic power is increased considerably at the optimum value of Cr . C. Effect of Cr on Tuning Range and Phase Noise We have shown that by using the proposed technique, the output power and efficiency of the push–push VCO clearly improve but just how does it affect the other two very

Fig. 11. Simulated load–pull power contours to find Z opt for the 0.3 THz VCO. (a) Conventional design without using Cr . The constant power contours are plotted from −12 to −6 dBm with a 1-dB step. Popt is −5 dBm. (b) Proposed design using an optimum value of Cr for common-mode impedance enhancement. The power contours are plotted from −10 to 0 dBm with a 1-dB step. Popt is +1.7 dBm.

important parameters of the VCO: the tuning range and the phase-noise performance. Ideally, Cr should not influence the output frequency; however, there is still a slight dependence of output frequency on Cr . It should be noted that for the entire tuning range of the VCO, Cr is kept constant and is implemented using an MIM capacitor. The optimal value of Cr is found at the center of the tuning range. In Fig. 13(a), we see that the tuning bandwidth of the VCO is decreased by around 2 GHz when Cr is employed, which corresponds to around 0.6% decrease in the normalized tuning range (FTR). To see how much the optimal value of Cr changes throughout the tuning range, the output power of the second harmonic is plotted for different tuning frequencies as a function of Cr .

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Fig. 12. Simulated plots of the VCO. (a) Voltage swing at the load R L , without and with an optimized value of Cr . (b) Fundamental and second harmonic output power at the load as a function of Cr .

Fig. 13. (a) Simulated tuning characteristics of the VCO without and with an optimized value of Cr . (b) Output power of the optimized VCO for each tuning frequency as a function of Cr .

For each tuning frequency, the value of Cr is swept and the output power is noted. We see that for the simulated tuning range, the optimal value of Cr changes by only 5 fF or less than 20%. The influence of Cr on the change in output frequency fo and on the phase noise is plotted in Fig. 14. The simulated output frequency changes by about 2 GHz (∼ 0.6%) if Cr is swept from 1 to 100 fF. This shows that once the VCO is designed for a particular f osc , introducing Cr will have a negligible effect on f o . The simulated phase noise at an offset of 10 MHz is plotted in Fig. 14 as a function of Cr , which shows a negligible change. D. Design of TL E and the VCO Layout The transmission line TL E provides a path for the biasing current to the virtual ground. The line’s length influences the oscillation frequency, output power, and the phase noise. Ideally, this line should present a very high impedance to the fundamental signal so that the large signal swing is confined and subsequently enhanced within the VCO core. Using large inductances consumes too much chip area and a better choice is to therefore use quarter-wave transmission lines (at fosc ). A properly designed length provides isolation from the capacitance of the current source Q 2 and desensitizes the tuning range from its capacitance. TL E together with the

Fig. 14. Frequency pulling and phase noise as a function of Cr at a tuning voltage of 3 V. The phase noise is plotted at an offset of 10 MHz.

output capacitance of Q 2 forms an LC filter, which decouples high-frequency noise, especially around 2 f osc . This noise can otherwise increase the phase noise considerably [62]. TL E is designed in such a way that its resonance with Cvar is less than the fundamental resonance of the VCO [63] 1 < f osc . 2π L TLE Cvar

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TABLE II PARAMETER VALUES OF VCO1 AND VCO2

Fig. 15. VCO2 core layout shows the via stack of the HBTs, MOS varactor, MIM capacitors, and the microstrip transmission lines implemented on TM2.

Table II provides the component values of the two designed VCOs (VCO1 and VCO2) with different HBT and varactor sizes. The layout of the core of VCO2 is shown in Fig. 15. The HBTs and the varactor are placed as close as possible in order to reduce footprint, interconnect inductance, and parasitic capacitance. The base terminals of the HBTs are contacted from the center to ensure minimum series resistance. The topmost metal (3 μm thick) is used for all the microstrip transmission lines. The collector terminals are shorted via M5, which directly connects to one of the MIM capacitor CC terminal. The layout is designed to be as symmetrical as possible. V. C HARACTERIZATION OF THE VCOs Micrographs of the fabricated VCO chips are shown in Fig. 16. Each VCO chip occupies an area of around 0.3 mm2 . As shown in the schematic (see Fig. 9), a bias voltage pad (VB ) has been included so that the behavior of output power, frequency, and phase noise as a function of bias current density can be observed. However, V B can be left unconnected for normal operation. The transmission lines indicated in the schematic of the VCO can be clearly seen in the close-up view of the micrograph in Fig. 16(b). All measurements were performed on-wafer at room temperature using WR-03 band GSG microwave probes from Cascade.

Fig. 16. (a) Micrograph of VCO1. Total chip area is 0.3 mm2 . (b) Close-up view of micrograph of VCO2. DC pads are not shown. The active chip area is around 0.05 mm2 .

Measurement setup for frequency tuning and phase noise is demonstrated in Fig. 17(a). A WR-03 S-bend of around 13 cm was required for an onward connection of the probe. A WR-03 band frequency converter from Rhode & Schwarz was used to downconvert the output signal into an IF signal in the range of 400 MHz. A Keysight 7-GHz signal source analyzer (SSA) E5052B was used to measure both the frequency and the phase-noise performance. The SSA provides a very low noise dc supply and tuning voltage and reduces the short-term and long-term signal phase instabilities. The tuning characteristics for VCO1 and VCO2 were measured at a constant supply voltage of 3.3 V and are shown in Fig. 18. The tuning curves follow the classic profile of an MOS varactor and show a sharp increase in the frequency up to 2 V, after which the capacitance of the varactor begins to saturate. The VCO2 demonstrates a frequency range from 292 to 318 GHz with a relative bandwidth of 8.5%. The VCO1 shows a maximum frequency of 327 GHz with a tuning bandwidth of 7.0%. Measurement setup for characterizing the output power is shown in Fig. 17(b). The output power was measured

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Fig. 17. On-wafer measurement setups for the VCO chips. (a) Frequency and phase noise measurements employing R & S frequency converters and Keysight SSA. (b) WR-03 band power measurement employing Erickson’s PM4 power meter.

Fig. 19. Measured output power of the VCO chips at different supply voltages as a function of the tuning voltage. (a) VCO1. (b) VCO2.

Fig. 18. Measured tuning characteristics of the VCO chips at a supply voltage of 3.3 V.

using Erickson PM4 calorimeter with a WR-10-based power sensor. A WR-03–WR-10 waveguide transition was therefore used in the setup. The losses associated with the probes and the waveguides were deembedded from the measurements. The measured loss of the probe as provided by the manufacturer is around 5 dB at 300 GHz. The loss of the interconnecting waveguide sections was measured to be 4.5 dB. The measured output power of the VCO chips is shown in Fig. 19. Since the biasing network is tied with VCC , raising the supply voltage increases the current density of the HBTs correspondingly. In regard to VCO1, the core current increases from 20, 26.5, 34, and 42 mA, as supply voltage is varied from 2.7 to 3.6 V in 0.3-V steps, respectively. Both VCO1 and VCO2 demonstrate the highest output power at a supply voltage of 3.3 V, which correspond to a current density of slightly higher than peak- f T . VCO1 shows a peak output power of 0.2 dBm with a dc-to-RF efficiency of 0.94%. However, it has a peak efficiency of 1% at a VCC of 3.0 V

with a peak power of −0.5 dBm. At a supply voltage of 3.3 V, it has a power variation of almost 1 dB over the entire tuning range, while the variation is within 2 dB for the other supply voltages. VCO2 demonstrates a higher peak output power of 0.6 dBm with a dc-to-RF efficiency of 0.8%. This is the highest output power attained for SiGe-based push–push tunable signal sources working above 270 GHz. Due to higher losses associated with varactors, single-frequency oscillators tend to achieve higher output power and efficiency [27]. The output power variation with tuning frequency for VCO2 is almost within 3 dB for all supply voltages. Phase noise measurements were performed using the setup shown in Fig. 17(a) at a supply voltage of 3.3 V. Fig. 20(a) shows the measured phase noise of the VCOs at an offset of 1 and 10 MHz with respect to the tuning voltage. The average phase noise at an offset of 1 MHz for VCO1 and VCO2 over the entire frequency range is −83.9 and −82.7 dBc/Hz, respectively. Similarly, at an offset of 10 MHz, the average phase noise is −102.1 and −103.2 dBc/Hz for VCO1 and VCO2, respectively. The phase-noise performance of both VCOs is improved at higher tuning voltages. This is predominantly due to the improved quality factor of the MOS varactor at higher reverse voltages. In full depletion, the relatively flat capacitance of the varactor reduces the phase noise

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TABLE III C OMPARISON W ITH S TATE - OF - THE -A RT S IGNAL S OURCES OVER 200 GHz

generated by the AM–FM conversion process [28]. Fig. 20(b) shows the phase noise as a function of offset frequency at a tuning voltage of 6 V. VCO2 achieves a minimum phase noise of around −108 dBc/Hz at 10 MHz, which is the lowest reported value for Si-based signal sources working above 260 GHz. Table III shows a comparison of state-ofthe-art signal sources working above 200 GHz. An figure-ofmerit (FoM) which includes FTR and Pout has been used for a more comprehensive performance comparison in Table III,

as given by [70]



 f o FTR FoMT = L( f ) − 20 log  f 10   Pdc +10 log (17) − POUT 1 mW where f o is the center frequency of the tuning range. As shown in Fig. 21, both VCO1 and VCO2 show the best FoMT for oscillator-based fixed and tunable signal sources working

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oscillators. The technique involves maximizing the commonmode impedance by introducing a parallel common-mode resonance at the second harmonic of the fundamental oscillation frequency. Although prior works have utilized commonmode impedance optimization to enhance the performance of harmonic oscillators, the presented approach is unique in terms of its easy design, simple implementation, and significant improvement in output power and efficiency while having almost negligible effect on tuning bandwidth. Based on the proposed technique, two voltage-controlled oscillators working beyond 0.3 THz are implemented in a 130-nm SiGe BiCMOS technology. As compared with a conventional Colpitts topology with the same dc power consumption, it is shown that the output power is improved by more than 6 dB. The fabricated chips outperform the conventional SiGe-based push–push VCOs working in this frequency range and show the best FoM for signal sources working above 270 GHz. R EFERENCES

Fig. 20. Measured phase-noise performance of the VCO chips at a supply voltage of 3.3 V. (a) Phase noise plot versus the tuning voltage. (b) Phase noise plot versus offset-frequency at a tuning voltage of 6 V.

Fig. 21. Comparison of FoMT of state-of-the-art signal sources in Si/SiGe and InP HBT technologies. This paper presented here represents the highest FoMT in its frequency range.

above 270 GHz. For frequency multiplier-based signal sources, the phase noise at the output depends upon the phase noise of the reference oscillator, the multiplication factor, and the additional noise added from the multiplier circuit. VI. C ONCLUSION We presented a new technique for improving the output power and dc-to-RF efficiency of push–push-based mm-wave

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[63] H. Li and H. M. Rein, “Millimeter-wave VCOs with wide tuning range and low phase noise, fully integrated in a SiGe bipolar production technology,” IEEE J. Solid-State Circuits, vol. 38, no. 2, pp. 184–191, Feb. 2003. [64] J.-Y. Kim, H.-J. Song, K. Ajito, M. Yaita, and N. Kukutsu, “A 325 GHz quadrature voltage controlled oscillator with superharmonic-coupling,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 8, pp. 430–432, Aug. 2013. [65] V. Radisic et al., “A 330-GHz MMIC oscillator module,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 395–398. [66] C. Jiang, A. Cathelin, and E. Afshari, “An efficient 210 GHz compact harmonic oscillator with 1.4 dBm peak output power and 10.6% tuning range in 130 nm BiCMOS,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., May 2016, pp. 194–197. [67] R. Han et al., “A SiGe terahertz heterodyne imaging transmitter with 3.3 mW radiated power and fully-integrated phase-locked loop,” IEEE J. Solid-State Circuits, vol. 50, no. 12, pp. 2935–2947, Dec. 2015. [68] H. Jalili and O. Momeni, “A 318-to-370GHz standing-wave 2D phased array in 0.13 μm BiCMOS,” in IEEE ISSCC Dig. Tech. Papers, Feb. 2017, pp. 310–312. [69] Y. Zhao et al., “A 0.56 THz phase-locked frequency synthesizer in 65 nm CMOS technology,” IEEE J. Solid-State Circuits, vol. 51, no. 12, pp. 3005–3019, Dec. 2016. [70] P.-Y. Chiang, O. Momeni, and P. Heydari, “A 200-GHz inductively tuned VCO with −7-dBm output power in 130-nm SiGe BiCMOS,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3666–3673, Oct. 2013. [71] A. Mostajeran and E. Afshari, “An ultra-wideband harmonic radiator with a tuning range of 62 GHz (28.3%) at 220 GHz,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Jun. 2017, pp. 164–167. [72] S. Jameson, E. Halpern, and E. Socher, “A 300 GHz wirelessly locked 2×3 array radiating 5.4 dBm with 5.1% DC-to-RF efficiency in 65 nm CMOS,” in IEEE ISSCC Dig. Tech. Papers, Jan./Feb. 2016, pp. 348–350. [73] J. Grzyb, B. Heinemann, and U. R. Pfeiffer, “Solid-state terahertz superresolution imaging device in 130-nm SiGe BiCMOS technology,” IEEE Trans. Microw. Theory Techn., to be published, doi: 10.1109/TMTT.2017.2684120. [74] N. Sarmah, P. R. Vazquez, J. Grzyb, W. Foerster, B. Heinemann, and U. R. Pfeiffer, “A wideband fully integrated SiGe chipset for high data rate communication at 240 GHz,” in Proc. Eur. Microw. Integr. Circuits Conf., Oct. 2016, pp. 181–184. [75] J. Grzyb, K. Statnikov, N. Sarmah, B. Heinemann, and U. R. Pfeiffer, “A 210–270-GHz circularly polarized FMCW radar with a single-lenscoupled SiGe HBT chip,” IEEE Trans. THz Sci. Technol., vol. 6, no. 6, pp. 771–783, Nov. 2016.

Faisal Ahmed (S’15) received the B.E. degree (Hons.) in electronic engineering from the NED University of Engineering and Technology, Karachi, Pakistan, in 2006, and the master’s degree (Hons.) in microwave engineering from the Technical University of Munich, Munich, Germany, in 2009. He is currently pursuing the Ph.D. degree at the Institute for Communications Engineering and RF-Systems, Johannes Kepler University, Linz, Austria. In 2006, he joined the Pakistan Space and Upper Atmosphere Research Commission (SUPARCO), where he was involved in on-board satellite telemetry and telecommand subsystems. From 2010 to 2011, he was a Manager (payload subsystem) with the China Academy of Space and Technology, involved in the development and launch of the Paksat-1R Communication Satellite. In 2017, he joined Danube Integrated Circuit Engineering, Linz, as a Millimeter-Wave Circuit Designer. His current research interests include analog-integrated circuit and system design for millimeter and terahertz waves for radar sensors and microwave imaging. Mr. Ahmed was the recipient of the Excellence Award in Satellite Research of SUPARCO in 2006 and 2011. He was also the recipient of the 2015 Best Student Contribution Award of the Radio Frequency Engineering Working Group (ARGE HFT) of the Austrian Research Association and the 2016 International Journal of Microwave and Wireless Technologies Best Paper Award.

Muhammad Furqan (S’12) received the B.E. degree (Hons.) in electronic engineering from the NED University of Engineering and Technology, Karachi, Pakistan, in 2006, and the master’s degree (Hons.) in microwave engineering from the Technical University of Munich, Munich, Germany, in 2009. He is currently pursuing the Ph.D. degree at the Institute for Communications Engineering and RF-Systems, Johannes Kepler University, Linz, Austria. He was with the Computational Electromagnetics Laboratory, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia, from 2010 to 2012, where he was involved in numerical techniques in electromagnetics with an emphasis on ferrite materials. In 2017, he joined Danube Integrated Circuit Engineering, Linz, Austria, as a MillimeterWave Circuit Designer. His current research interests include circuit and system design for millimeter-wave communication systems and microwave radar sensors. Mr. Furqan was the recipient of an Academic Award for Excellence in 2010 and 2011. He was also the recipient of the 2016 International Journal of Microwave and Wireless Technologies Best Paper Award. He was honorably mentioned at the 2017 IEEE MTT-S International Microwave Symposium Student Paper Competition. Bernd Heinemann received the Diploma degree in physics from the Humboldt Universität zu Berlin, Berlin, Germany, in 1984, and the Ph.D. degree in electrical engineering from the Technische Universität Berlin, Berlin, in 1997. In 1984, he joined Innovations for High Performance Microelectronics, Frankfurt (Oder), Germany. From 1984 to 1992, he contributed to the development of an epi-free 0.8-μm BiCMOS technology. Since 1993, he has been a member of a team involved with the exploration and technological implementation of SiGe HBTs. His current research interests include the development and characterization of MOS and bipolar devices. Andreas Stelzer (M’00) was born in Haslach an der Mühl, Austria, in 1968. He received the Dipl.Ing. degree in electrical engineering from the Technical University of Vienna, Vienna, Austria, in 1994, and the Dr. Techn. (Ph.D.) degree (Hons.) in mechatronics from Johannes Kepler University, Linz, Austria, in 2000. In 2003, he joined the Institute for Communications Engineering and RF-Systems, Johannes Kepler University, as an Associate Professor. He has been a Key Researcher with the Austrian Center of Competence in Mechatronics, Linz, since 2008, where is currently responsible for numerous industrial projects. He has been the Head of the Christian Doppler Research Laboratory for Integrated Radar Sensors, since 2007, and a Full Professor with Johannes Kepler University, since 2011, heading the Department of RF-Systems. He has authored or co-authored over 360 journal and conference papers. His current research interests include microwave sensor systems for industrial and automotive applications, radar concepts, SiGe-based circuit design currently up to 320 GHz, microwave packaging in eWLB, RF, and microwave subsystems, surface acoustic wave sensor systems and applications, and digital signal processing for sensor signal evaluation. Dr. Stelzer is a member of the Austrian Electrotechnical Association, the IEEE Microwave Theory and Techniques Society (MTT-S), the IEEE Instrumentation and Measurement Society, and the IEEE Circuits and Systems Society. He serves as the IEEE Distinguished Microwave Lecturer from 2014 to 2016. He was the recipient of several awards including the 2008 IEEE MTT-S Outstanding Young Engineer Award, the 2011 IEEE Microwave Prize, the 2012 European Conference on Antennas and Propagation Best Measurement Paper Prize, the 2012 Asia–Pacific Conference on Antennas and Propagation Best Paper Award, the 2011 German Microwave Conference Best Paper Award, the EEEfCOM Innovation Award, and the 2003 European Microwave Association Radar Prize of the European Radar Conference. He has served as an Associate Editor for IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS . He currently serves as the Co-Chair for MTT-27 Wireless-Enabled Automotive and Vehicular Applications.

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1

Inkjet Printing of Epidermal RFID Antennas by Self-Sintering Conductive Ink S. Amendola , A. Palombi, and G. Marrocco

Abstract— The recently introduced inkjet printing technology with ambient sintering is here investigated for the fabrication of epidermal antennas suitable for radio-frequency identification (RFID) and sensing. The attractive feature of this manufacturing process is the possibility to use low-cost printers without any high-temperature curing. In spite of the estimated maximum achievable conductivity of the ink (σUHF = 1 × 105 S/m) in the UHF-RFID band that is two orders of magnitude lower than that of the bulk copper, a threefold printing process provides the same on-skin radiating performance as manufacturing technologies using bulk conductors. Experiments demonstrate that the epidermal antennas printed on the PET substrate are insensitive to moderate mechanical stress, like the natural bending occurring over the human body, and to the possible exposure to body fluids (e.g., sweat). Moreover, the electromagnetic response remains stable over the time when the printed layouts are coated with biocompatible membranes. Index Terms— Additive manufacturing, body sensor network, inkjet printing, radio-frequency identification (RFID), RFID tags, UHF antennas, wearable sensor, wireless sensor.

I. I NTRODUCTION

F

LEXIBLE and body-conformable sensors are a promising driver for the new generation of noninvasive and discrete body-centric systems with application to biomedicine, security, and entertainment. The recent convergence between the emerging Epidermal Electronics [1] and the more assessed radio-frequency identification (RFID) technology for passive body-centric systems [2] is indeed stimulating the development of novel skin-tight batteryless devices provided with sensor capabilities and wireless interfaces for communication with a remote reader unit. Pioneering applications of epidermal RFID tags to the wireless measurement of body temperature and to wound healing monitoring were already demonstrated in [3] and [4]. Reshaping RFID transponders, conventionally used in logistics of bulk objects, into a suitable layout for skin mounting demands for techniques to deposit conductive traces over biocompatible ultrathin flexible membranes. Cost-effective and easily accessible methods are, hence, required for both the rapid prototyping of laboratory samples as well as for the mass production of skin sensors over the large scale. The first Manuscript received July 13, 2017; revised September 27, 2017; accepted October 4, 2017. (Corresponding author: S. Amendola.) The authors are with the Pervasive Elecromagnetics Laboratory, University of Rome Tor Vergata, 00173 Rome, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2767594

referred prototypes of tattoo-like tags, in the UHF RFID band (860–960 MHz), were fabricated by profiling conductive silver painting or a nickel-based screening spray by means of the stencil technique [5]. Inkjet printing is a promising technology for depositing metal traces on flexible and even stretchable substrates [6]–[8]. This process is being increasingly adopted by the electronics industry for the fabrication of RF circuits and wireless devices [9] by using electrically engineered inks made by metallic nanoparticle, conductive polymers (PEDOT-PSS), organometallic compounds, and carbon nanotubes (refer to [10] and [11] for a complete survey). Inkjet printing of silver nanoparticle inks has already been experimented also for manufacturing RFID tags [12], [13] and even to produce epidermal antennas over transfer tattoo paper [14]. Most of the published works refer to highly specialized and expensive printing equipment, like the FUJIFILM Dimatix DMP-2800, with nanoparticle silver-based inks (e.g., from Xerox, Sigma-Aldrich). This technology requires thermal or laser postdeposition sintering treatments at high temperature (between 135° and more than 300°) to provoke the coalescence of the nanoparticles that are enclosed in a polymeric shell—especifically designed to forbid agglomeration prior to the deposition—and, accordingly, to achieve the optimal electrical conductivity. Depending on the specific printing process, the kind of substrate, the number of the printed layers, and the resulting thickness of the traces, the dc conductivity of printed ink nowadays approaches the same order of magnitude (σ ÷ 107 S/m) as bulk conductors [σ (Ag) = 6.3 × 107 S/m, σ (Cu) = 5.9 × 107 S/m, and σ (Al) = 3.5 × 107 S/m]. Just to give an example, a conductivity of σ = 2 × 107 S/m was measured at 2450 MHz in [15] for a silver-based ink printed on cardboard and thermally cured. Very recent progresses in materials science originated a new class of conducting inks [16], [17] that dry at room temperature and form an instantly conductive layer, without the need of time-consuming thermal sintering [18], [19]. These sintering-free inks are, hence, suitable to be cheaply and easily deposited on a flexible substrate by using consumer-grade, low-cost, inkjet printers. The feasibility of the fabrication of a variety of functional electronic prototypes, including touch and proximity-sensitive surfaces and capacitive liquid-level sensors, has already been demonstrated in [20] and [21]. Ta et al. [22] proposed procedures to make interconnections between double-sided patterns with the purpose of fabricating multilayered instant-printed circuits. Even more recently,

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the self-sintering ink was used for the fabrication of miniaturized artificial magnetic conductors in the sub-GHz applications [23] and for the manufacturing of chipless humidity RFID sensors [24]. The declared spatial resolution of the printed trace was 150 μm while the ink sheet resistance was estimated to be 0.3 /m (with no indication about the number of printed layers). The application of the above manufacturing technique to epidermal antennas working in the UHF-RFID band requires a more in-depth analysis to take care of a number of additional parameters that are specific to the cohabitation of an antenna with the human skin, such as the printing quality over common biocompatible substrates, the presence of the sweat, the possible bending over body curvatures, and the need of a protective coating. In spite of some information about the radio-frequency feature of the self-sintering inkjet printing of antennas that may be derived from the above cited papers, other issues, such as the achievable conductivity versus the specific inkjet process (say the number of printed layers), the long-term performance, and the robustness to the mechanical and chemical stress, are still unknown and worthwhile for further investigation. Thus, this paper describes a complete and independent characterization of the self-sintering inkjet technology for the specific fabrication of epidermal antennas with the overall goal of identifying the most appropriate modalities to achieve readrange performance as close as possible to the bulk copper in the case of application onto the skin, and to test the sensitivity of the inkjet-printed antennas in variable boundary conditions typical of body-centric systems. The basics of the sinteringfree conductive ink are reviewed in Section II regarding the ink conductivity in the dc regime and the identification of suitable printing substrates. Then, in Section III, the ink conductivity is estimated in the UHF-RFID band by means of a combined experimental/simulated identification procedure versus the number of printed layers. Section IV addresses the achievable performance of realistic epide